Answer:
Step-by-step explanation:
Given that X, the mutual fund returns in the 1st quarter of 2013, is
N(0.054, 0.015)
a) P(X>6.8%) = [tex]1-0.8246\\=0.1754[/tex]
b) [tex]P(0<X<0.076)\\\\\\=0.9288-0.0002\\=0.9286[/tex]
c) [tex]P(X>0.01)\\=1-0.0017\\=0.9983[/tex]
d) [tex]P(X<0) = 0.0002[/tex]
A) The expected percentage of returns that are over 6.8% is _17.5_____%
b) The expected percentage of returns that are between 0& and 7.6% is __92.9__%
C) The expected percentage of returns that are more than 1% is _99.8___%
D) The expected percentage of returns that are less than 0% is _0.02___%
Red and white paint are mixed to make pink paint. If the amount of red paint needed is half the amount of white paint, how many gallons of red paint are needed to make 10 gallons of pink paint?
Answer:
We will need about 3.3 gallons of red paint.
Step-by-step explanation:
Let
r = amount of red paintw = amount of white paintThe amount of red paint is half the amount of white paint.
r = 1/2 w
w = 2r [1]
We need to make a total of 10 gallons. Then,
r + w = 10 [2]
Replacing [1] in [2]
r + (2r) = 10
3r = 10
r = 3.3
We will need about 3.3 gallons of red paint.
To make 10 gallons of pink paint with red paint being half the amount of white paint, you would need 5 gallons of red paint.
Explanation:The amount of red paint needed to make 10 gallons of pink paint when the amount of red paint is half that of white paint:
Let the amount of white paint be 'x', then the amount of red paint needed is 'x/2'.Given that white paint is 10 gallons, so 'x = 10'.Therefore, the amount of red paint needed is '10/2 = 5 gallons'.Integrated circuits consist of electric channels that are etched onto silicon wafers. A certain proportion of circuits are defective because of "undercutting," which occurs when too much material is etched away so that the channels, which consist of the unetched portions of the wafers, are too narrow. A redesigned process, involving lower pressure in the etching chamber, is being investigated. The goal is to reduce the rate of undercutting to less than 5%. Out of the first 1000 circuits manufactured by the new process, only 33 show evidence of undercutting. Can you conclude that the goal has been met? Find the P-value and state a conclusion.
Answer:
[tex]z=\frac{0.033 -0.05}{\sqrt{\frac{0.05(1-0.05)}{1000}}}=-2.467[/tex]
[tex]p_v =P(Z>-2.467)=0.0068[/tex]
So the p value obtained was a very low value and using the significance level assumed for example [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of circuits that show evidence of undercutting is significantly less than 0.05.
Step-by-step explanation:
1) Data given and notation
n=1000 represent the random sample taken
X=33 represent the number of circuits that show evidence of undercutting
[tex]\hat p=\frac{33}{1000}=0.033[/tex] estimated proportion of circuits that show evidence of undercutting
[tex]p_o=0.05[/tex] is the value that we want to test
[tex]\alpha[/tex] represent the significance level
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
2) Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that they reduce the rate of undercutting to less than 5%.:
Null hypothesis:[tex]p\geq 0.05[/tex]
Alternative hypothesis:[tex]p < 0.05[/tex]
When we conduct a proportion test we need to use the z statistic, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
3) Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.033 -0.05}{\sqrt{\frac{0.05(1-0.05)}{1000}}}=-2.467[/tex]
4) Statistical decision
It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.
The next step would be calculate the p value for this test.
Since is a left tailed test the p value would be:
[tex]p_v =P(Z>-2.467)=0.0068[/tex]
So the p value obtained was a very low value and using the significance level assumed for example [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of circuits that show evidence of undercutting is significantly less than 0.05.
Let U represent the set of people in a certain community who were asked if they subscribe to an information source.
Let D ={ x ∈ U | x subscribes to The Daily Informer }
and
N ={ x ∈ U | x subscribes to News Magazine }
(Assume these sets are not disjoint.) Write the set that represents the set of people surveyed who subscribe to exactly one of the two news sources given.
a) N ∪ D
b) ( N c ∩ D ) ∪ ( N ∩ D c )
c) N c ∩ D
d) ( N ∩ D )c
e) ( N c ∪ D ) ∩ ( N ∪ D c )
Answer:
b) ( N c ∩ D ) ∪ ( N ∩ D c )
Step-by-step explanation:
Let's see each option:
a) N ∪ D
This includes those who likes both, so this is not correc.t
b) ( N c ∩ D ) ∪ ( N ∩ D c )
Nc is the complement of N. That is those who do not subscribe to news magazine. Intensected with D, those are who do not subscribe to the news magazine but do to the daily informer. By the same logic, the second intersection are those who do subscribe to news magazine but not to the daily informer. This is the correct answer
c) N c ∩ D
This includes only those who subscribe to the daily informer but not to the news maganize. It also needs those who subscribe to the news magazine but not to the daily informer.
d) ( N ∩ D )c
This also involves those who do not subscribe to any of these news sources.
e) ( N c ∪ D ) ∩ ( N ∪ D c )
Wrong... Intersection of those who subscribe to the daily informer but not the news magazine and those who subscribe to the news magazine but not to the daily informer. This is the empty set.
A tourist in Ireland wants to visit six different cities. How many different routes are possible?
A. 46,656
B. 120
C. 36
D. 720
Answer:
[tex] 6P6 = \frac{6!}{(6-6)!}=\frac{6!}{0!}= 6! =6*5*4*3*2*1=720[/tex]
D. 720
Step-by-step explanation:
For this case we have 6 different cities that needs to be ordered in different ways. And the best way to solve this problem is using permutations since we can't repeat the route.
We need to remember the concept of permutation.
A permutation, known as "arrangement number" "is a rearrangement of the elements of an ordered list S into a one-to-one correspondence with S itself". Where S correspond to the sample space. And the formula is given by:
[tex]nPX = \frac{n!}{(n-x)!}[/tex]
And for this case we have a total of 6 cities and we want to know how many routes with these 6 cities we can create, so then n=6 and k=6 and if we replace we got:
[tex] 6P6 = \frac{6!}{(6-6)!}=\frac{6!}{0!}= 6! =6*5*4*3*2*1=720[/tex]
And then we have 720 ways to visit six different cities. So the best option is:
D. 720
The number of different routes possible for a tourist visiting six cities in Ireland is 720, calculated by the formula for permutations of a set (6!).
Explanation:The question is asking for how many different routes are possible if a tourist wants to visit six different cities in Ireland. This is an application of the mathematical principle of permutations, which is a way of counting or ordering objects. In this case, for 6 cities, we use the formula for permutations of a set, which is represented as n!, where n is the number of cities to visit.
For 6 cities, this would be 6!, which is calculated as 6 * 5 * 4 * 3 * 2 * 1 = 720. So, there are 720 different routes that the tourist can take to visit the six different cities.
Hence, option 'D. 720' is the correct answer.
Learn more about Permutations here:https://brainly.com/question/23283166
#SPJ3
Determine which of the following show three biased estimators. (1 point) a. median, mean, range b. range, standard deviation, variance c. standard deviation, median, ranged. variance, proportion, mean
Answer:
c. standard deviation, median, range
Step-by-step explanation:
The standard deviation without the Bessel's correct is defined as:
[tex] s= \sqrt{\frac{\sum_{i=1}^n (x_i -\bar x)^2}{n}[/tex]
And if we find the expected value for s we got:
[tex] E(s^2) = \frac{1}{n} \sum_{i=1}^n E(x_i -\bar x)^2 [/tex]
[tex] E(s^2)= \frac{1}{n} E[\sum_{i=1}^n ((x_i -\mu)-(\bar x -\mu)^2)][/tex]
We have this:
[tex] E(\sum_{i=1}^n(x_i-\mu)^2) =n\sigma^2[/tex]
[tex]E[\sum_{i=1}^n (x_i -\mu)(\bar x -\mu)]= \sigma^2[/tex]
[tex]E[\sum_{i=1}^n (\bar x -\mu)^2]=\sigma^2[/tex]
[tex]E(s^2)=\frac{1}{n} (n\sigma^2 -2\sigma^2 +\sigma^2)[/tex]
[tex]E(s^2)=\frac{n-1}{n}\sigma^2[/tex]
as we can see the sample variance is a biased estimator since:
[tex]E(s^2)\neq \sigma^2[/tex]
And we see that the standard deviation is biased, since:
[tex] E(s) = \sqrt{\frac{n-1}{n}} \sigma[/tex]
because [tex]E(s)\neq \sigma[/tex]
The mean is not biased for this case option a is FALSE.
The proportion is not biased for this reason option d is FALSE
The range can be considered as biased since we don't have info to conclude that the range follows a distirbution in specific.
The sample median "is an unbiased estimator of the population median when the population is normal. However, for a general population it is not true that the sample median is an unbiased estimator of the population median".
And for this reason the best option is c.
Answer: standard deviation, median, range
Step-by-step explanation:
A recent study by Ohio State University reported at Science Daily suggests that students with cell phones may take more risks than students that do not have cell phones. In a sample of 305 Ohio State University female students, 128 (42%) responded that if they had a cell phone, they would be willing to walk somewhere after dark that they would normally not go.
Use the above survey results to test the hypotheses
H0: p = 0.50
HA: p < 0.50
where p is the proportion of female students who, if they had a cell phone, would be willing to walk somewhere after dark that they would normally not go.
Question 1. What is the value of the test statistic z for this hypothesis test? (Use 2 decimal places in your answer).
Answer:
The value of the test statistic z for this hypothesis test is -2.79
Step-by-step explanation:
Consider the provided information.
To calculate the test statistic use the formula:
[tex]z=\frac{\hat p-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}[/tex]
Where, z is Test statistics, n is Sample size, p₀ = Null hypothesized value and [tex]\hat p[/tex] = Observed proportion.
p₀ = 0.50
Thus 1-p₀= 0.50
42% responded that if they had a cell phone, thus [tex]\hat p=0.42[/tex]
The sample size is 305.
Substitute the respective values in the above formula.
[tex]z=\frac{0.42-0.50}{\sqrt{\frac{0.50(0.50)}{305}}}[/tex]
[tex]z=\frac{-0.08}{\sqrt{0.00082}}[/tex]
[tex]z=-2.79[/tex]
Hence, the value of the test statistic z for this hypothesis test is -2.79
The value of the test statistic [tex]\( z \)[/tex] for this hypothesis test is approximately -2.797.
To calculate the test statistic [tex]\( z \)[/tex], we follow these steps:
1. State the null hypothesis [tex]\( H_0: p = 0.50 \)[/tex] and the alternative hypothesis [tex]\( H_A: p < 0.50 \)[/tex].2. Identify the sample proportion [tex]\( \hat{p} \)[/tex], which is the proportion of the sample that has the characteristic of interest.
In this case, [tex]\( \hat{p} = \frac{128}{305} \approx 0.42 \)[/tex]
3. Calculate the standard error of the proportion, which is given by [tex]\( SE = \sqrt{\frac{p(1-p)}{n}} \)[/tex], where [tex]\( p \)[/tex] is the population proportion under the null hypothesis and [tex]\( n \)[/tex] is the sample size
[tex]\( SE = \sqrt{\frac{0.50(1-0.50)}{305}} \approx \sqrt{\frac{0.25}{305}} \)[/tex]
4. Compute the test statistic [tex]\( z \)[/tex] using the formula [tex]\( z = \frac{\hat{p} - p}{SE} \)[/tex], where [tex]\( \hat{p} \)[/tex] is the sample proportion, [tex]\( p \)[/tex] is the population proportion under the null hypothesis, and [tex]\( SE \)[/tex] is the standard error.
5. Plug in the values to get [tex]\( z = \frac{0.42 - 0.50}{\sqrt{\frac{0.25}{305}}} \)[/tex].
6. Simplify the expression to find the value of [tex]\( z \)[/tex].
[tex]\[ z = \frac{0.42 - 0.50}{\sqrt{\frac{0.25}{305}}} = \frac{-0.08}{\sqrt{\frac{1}{1220}}} = \frac{-0.08}{\sqrt{0.0008197}} = \frac{-0.08}{0.0286} \approx -2.797 \][/tex]
Find the product. If the result is negative, enter "-". If the result is positive, enter "+". -7(- a2 ) 2 ( -b3 ).
Answer:
The product is positive, thus it is [tex]\bold{+7a^4b^3}[/tex]
Step-by-step explanation:
The full question in proper notation is:
"Find the product. If the result is negative, enter "-". If the result is positive, enter "+".
[tex]-7(-a^2)^2(-b^3)[/tex]"
We have to work with it using Order of operations know as well as PEMDAS, thus expression inside parenthesis go first and exponents.
On this expression we have to work with exponents
[tex](-a^2)^2 = (-a^2)(-a^2) =a^4[/tex]
Thus we get
[tex]-7(-a^2)^2(-b^3)=-7a^4(-b^3)[/tex]
Lastly we can work with multiplication and remembering that the multiplication of two negative signs becomes positive.
[tex]-7(-a^2)^2(-b^3)=7a^4b^3[/tex]
So the final simplified expression is [tex]\bold{7a^4b^3}[/tex]
Answer: +7a^4b^3
Step-by-step explanation:
What is the equation of a horizontal line that goes through the coordinate (4,-2)
The equation will be:
y = -2
Step-by-step explanation:
A horizontal line has no slope as it is parallel to x-axis.
The horizontal line is in the form y = b where b is the y-intercept.
Given
The line passes through (4 -2).
The y-coordinate of the given point is -2 which means that the line will intersect the y-axis on -2
So,
b = -2
and
The equation will be:
y = -2
Keywords: Equation of line, slope
Learn more about equation of line at:
brainly.com/question/6569012brainly.com/question/6541303#LearnwithBrainly
Suppose μ1 and μ2 are true mean stopping distances at 50 mph for cars of a certain type equipped with two different types of braking systems. The data follows: m = 5, x = 114.1, s1 = 5.08, n = 5, y = 129.9, and s2 = 5.37. Calculate a 95% CI for the difference between true average stopping distances for cars equipped with system 1 and cars equipped with system 2. (Round your answers to two decimal places.)
Answer:
The 95% confidence interval would be given by [tex]-23.44 \leq \mu_1 -\mu_2 \leq -8.16[/tex]
Step-by-step explanation:
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
[tex]\bar X_1 =114.1[/tex] represent the sample mean 1
[tex]\bar X_2 =129.9[/tex] represent the sample mean 2
n1=5 represent the sample 1 size
n2=2 represent the sample 2 size
[tex]s_1 =5.08[/tex] sample standard deviation for sample 1
[tex]s_2 =5.37[/tex] sample standard deviation for sample 2
[tex]\mu_1 -\mu_2[/tex] parameter of interest.
Confidence interval
The confidence interval for the difference of means is given by the following formula:
[tex](\bar X_1 -\bar X_2) \pm t_{\alpha/2}\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}[/tex] (1)
The point of estimate for [tex]\mu_1 -\mu_2[/tex] is just given by:
[tex]\bar X_1 -\bar X_2 =114.1-129.9=-15.8[/tex]
In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:
[tex]df=n_1 +n_2 -1=5+5-2=8[/tex]
Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,8)".And we see that [tex]t_{\alpha/2}=\pm 2.31[/tex]
The standard error is given by the following formula:
[tex]SE=\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}[/tex]
And replacing we have:
[tex]SE=\sqrt{\frac{5.08^2}{5}+\frac{5.37^2}{5}}=3.306[/tex]
Confidence interval
Now we have everything in order to replace into formula (1):
[tex]-15.8-2.31\sqrt{\frac{5.08^2}{5}+\frac{5.37^2}{5}}=-23.437[/tex]
[tex]-15.8+2.31\sqrt{\frac{5.08^2}{5}+\frac{5.37^2}{5}}=-8.163[/tex]
So on this case the 95% confidence interval would be given by [tex]-23.44 \leq \mu_1 -\mu_2 \leq -8.16[/tex]
To calculate a 95% confidence interval for the difference between the true average stopping distances for cars equipped with system 1 and system 2, use the formula CI = (x1 - x2) ± (c * SE) where x1 and x2 are the sample means, and SE is the standard error of the difference.
Explanation:To calculate a 95% confidence interval for the difference between the true average stopping distances for cars equipped with system 1 and system 2, we can use the formula:
CI = (x1 - x2) ± (c * SE)
where x1 and x2 are the sample means, and SE is the standard error of the difference. The critical value c can be found using the t-distribution table for the given sample sizes n1 and n2.
https://brainly.com/question/34700241
#SPJ3
Computer Help Hot Line receives, on average, 14 calls per hour asking for assistance. Assume the variable follows a Poisson distribution. What is the probability that the company will receive more than 20 calls per hour? Round answer to 4 decimal places.
Answer: 0.0479
Step-by-step explanation:
Given : Computer Help Hot Line receives, on average, 14 calls per hour asking for assistance.
Let x be number of variable that denotes the number of calls that follows a Poisson distribution.
Poisson distribution formula : [tex]P(X=x)=\dfrac{e^{-\lambda}\lambda^x}{x!}[/tex]
, where [tex]\lambda[/tex] =Mean of the distribution.
Here ,
Then, the probability that the company will receive more than 20 calls per hour= [tex]P(x>20)=1-P(x\leq20)[/tex]
[tex]=1-0.9521=0.0479 [/tex]
(From Cumulative Poisson distribution table the value of P(x ≤ 20) =0.9521 corresponding to [tex]\lambda=14[/tex] ).
Thus , the probability that the company will receive more than 20 calls per hour = 0.0479
An article reported on a school district's magnet school programs. Of the 18701870 qualified applicants, 963963 were accepted, 271271 were waitlisted, and 636636 were turned away for lack of space. Find the relative frequency for each decision made, and write a sentence summarizing the results.
To summarize the results, approximately 51.5% of the qualified applicants were accepted into the magnet school programs, around 14.5% were waitlisted, and about 34% were turned away due to lack of space.
Given that;
An article reported on a school district's magnet school programs.
Of the 18701870 qualified applicants, 963963 were accepted, 271271 were waitlisted, and 636636 were turned away for lack of space.
Let's calculate the relative frequency for each decision made.
To find the relative frequency, we divide the number of applicants by the total number of qualified applicants.
For the accepted applicants:
Relative frequency = Number of accepted applicants / Total qualified applicants
Relative frequency = 963 / 1870
Relative frequency ≈ 0.515
For the waitlisted applicants:
Relative frequency = Number of waitlisted applicants / Total qualified applicants
Relative frequency = 271 / 1870
Relative frequency ≈ 0.145
For the applicants turned away:
Relative frequency = Number of turned away applicants / Total qualified applicants
Relative frequency = 636 / 1870
Relative frequency ≈ 0.340
Thus, To summarize the results, approximately 51.5% of the qualified applicants were accepted into the magnet school programs, around 14.5% were waitlisted, and about 34% were turned away due to lack of space.
To learn more about the divide visit:
https://brainly.com/question/28119824
#SPJ12
A company with a large fleet of cars wants to study the gasoline usage. They check the gasoline usage for 50 company trips chosen at random, finding a mean of 27.02 mpg and sample standard deviation is 5.83 mpg. d. Please use R to construct a (two-sided) 88% CI for the mean of the general gasoline usage. Then for this answer, provide the lower bound of the CI and round to 2 decimal places. Please do not use the automagic R function. Only use functions that we've covered in class (or else you won't get credit).
Answer:
The 95% confidence interval is given by (25.71536 ;28.32464)
And if we need to round we can use the following excel code:
round(lower,2)
[1] 25.72
round(upper,2)
[1] 28.32
And the interval would be (25.72; 28.32)
Step-by-step explanation:
Notation and definitions
n=50 represent the sample size
[tex]\bar X= 27.2[/tex] represent the sample mean
[tex]s=5.83[/tex] represent the sample standard deviation
m represent the margin of error
Confidence =88% or 0.88
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
Calculate the critical value tc
In order to find the critical value is important to mention that we don't know about the population standard deviation, so on this case we need to use the t distribution. Since our interval is at 88% of confidence, our significance level would be given by [tex]\alpha=1-0.88=0.12[/tex] and [tex]\alpha/2 =0.06[/tex]. The degrees of freedom are given by:
[tex]df=n-1=50-1=49[/tex]
We can find the critical values in R using the following formulas:
qt(0.06,49)
[1] -1.582366
qt(1-0.06,49)
[1] 1.582366
The critical value [tex]tc=\pm 1.582366[/tex]
Calculate the margin of error (m)
The margin of error for the sample mean is given by this formula:
[tex]m=t_c \frac{s}{\sqrt{n}}[/tex]
[tex]m=1.582366 \frac{5.83}{\sqrt{50}}=14.613[/tex]
With R we can do this:
m=1.582366*(5.83/sqrt(50))
m
[1] 1.304639
Calculate the confidence interval
The interval for the mean is given by this formula:
[tex]\bar X \pm t_{c} \frac{s}{\sqrt{n}}[/tex]
And calculating the limits we got:
[tex]27.02 - 1.582366 \frac{5.83}{\sqrt{50}}=25.715[/tex]
[tex]27.02 + 1.582366 \frac{5.83}{\sqrt{50}}=28.325[/tex]
Using R the code is:
lower=27.02-m;lower
[1] 25.71536
upper=27.02+m;upper
[1] 28.32464
The 95% confidence interval is given by (25.71536 ;28.32464)
And if we need to round we can use the following excel code:
round(lower,2)
[1] 25.72
round(upper,2)
[1] 28.32
And the interval would be (25.72; 28.32)
You may believe that the gender of a salesperson influences the sales of cars. The best way to incorporate this predictor is by Group of answer choices
a. None of these answers are correct.
b. Using a single dummy variable in a regression model
c. Running two separate regressions, one for females and one for males.
d. Using two dummy variables in a regression model
Answer: The correct option is (b)
Using a single dummy variable in a regression model
Step-by-step explanation:
A regression model is a model that measures the relationship between a dependent variable and one or more independent variables. In this question the dependent variable y is the sales of the car and the independent variable is the choice of the gender of a salesperson, Which is single dummy variable.
From the half-life for 14C decay, 5715 yr, determine the age of the artifact
The age of the artifact is approximately 3378 years.
To determine the age of the artifact, we can use the concept of radioactive decay and the half-life of carbon-14 (14C), which is 5715 years.
The formula for the decay of a radioactive substance is given by:
[tex]A = A_0 \times (0.5)^{(t / T_{1/2})[/tex]
where:
A = Final activity of the sample (in counts per minute, cpm)
A₀ = Initial activity of the sample (in counts per minute, cpm)
t = Time elapsed (in years)
T₁/₂ = Half-life of the radioactive substance (in years)
We are given:
A₀ (activity of the artifact) = 38.0 cpm
A (activity of the standard of zero age) = 58.2 cpm
T₁/₂ (half-life of carbon-14) = 5715 years
Let's rearrange the formula to solve for t (age of the artifact):
[tex]t = T_{1/2} \times log_2(A / A_0)[/tex]
Now, plug in the given values:
t = 5715 × log₂(58.2 / 38.0)
Using a calculator:
t ≈ 5715 × log₂(1.53263158)
t ≈ 5715 × 0.591170846
t ≈ 3378.115647
Hence, the age of the artifact is approximately 3378 years.
Learn more about Half-Life click;
https://brainly.com/question/31666695
#SPJ12
As a result of radioactive decay, heat is generated uniformly throughout the interior of the earth at a rate of around 30 watts per cubic kilometer. (A watt is a rate of heat production.) The heat then flows to the earth's surface where it is lost to space. Let F (x,y,z) denote the rate of flow of heat measured in watts per square kilometer. By definition, the flux of F across a surface is the quantity of heat flowing through the surface per unit of time.
Answer:
a) [tex]div F = 27 \frac{W}{km^3}[/tex]
b) [tex]\alpha=\frac{27 W/km^3}{3}= 9\frac{W}{Km^3}[/tex]
c) [tex]T(0,0,0)=-\frac{10}{2(27000)} (0) +7605.185=7605.185[/tex]
Step-by-step explanation:
(a) Suppose that the actual heat generation is 27W/km3 What is the value of div F? div F- Include units)
For this case the value for div F correspond to the generation of heat.
[tex]div F = 27 \frac{W}{km^3}[/tex]
(b) Assume the heat flows outward symmetrically. Verify that [tex] F= \alpha r[/tex] where [tex]r=xi +yj+zk[/tex]. Find a α, (Include units.)
For this case we can satisfy this condition:
[tex]div[\alpha (xi +yj +z k)]]=\alpha(1+1+1)=3\alpha[/tex]
And since we have the value for the [tex]div F[/tex] we can find the value of [tex]\alpha[/tex] like this:
[tex]\alpha=\frac{27 W/km^3}{3}= 9\frac{W}{Km^3}[/tex]
(c) Let T (x,y,z) denote the temperature inside the earth. Heat flows according to the equation F= -k grad T where k is a constant. If T is in °C then k=27000 C/km. Assuming the earth is a sphere with radius 6400 km and surface temperature 20°C, what is the temperature at the center? 27,0 C/km.
For this case we have this:
[tex] F =-k grad T[/tex]
And grad T represent the direction of the greatest decrease related to the temperature.
So we have this equation:
[tex] 10(xi +yj+zk)=-27000 grad T[/tex]
And we can solve for grad T like this:
[tex] grad T = -\frac{10}{(27000)} (xi+yj+zk)[/tex]
Andif we integrate in order so remove the gradient on both sides we got:
[tex]T=-\frac{10}{2(27000)} (x^2 +y^2 +z^2) +C[/tex]
For our case we have the following condition:
[tex]x^2 +y^2 +z^2 = 6400 , T=20 C[/tex]
[tex] T=-\frac{1}{54000} (6400^2)+C =20[/tex]
And we can solve for C like this:
[tex] C= 20+\frac{6400^2}{5400}= 7605.185 [/tex]
So then our equation would be given by:
[tex]T=-\frac{10}{2(27000)} (x^2 +y^2 +z^2) +7605.185[/tex]
And for our case at the center we have that [tex]x^2+ y^2+ z^2 =0[/tex]
And we got:
[tex]T(0,0,0)=-\frac{10}{2(27000)} (0) +7605.185=7605.185[/tex]
A mile is equal to 5280 feet.if the highway department places a reflector every 25 feet how many reflectors will there be in 1 mile of highway
Answer:there will be 211 reflectors on 1 mile of highway
Step-by-step explanation:
A mile is equal to 5280 feet. if the highway department places a reflector every 25 feet, the number of reflectors that would be in 1 mile of the highway would be
Total number of feets / 25. It becomes. 5280/25 = 211.2 reflectors. Approximating 211.2, it becomes 211 reflectors.
Olanda needs for a future project. She can invest now at an annual rate of , compounded monthly. Assuming that no withdrawals are made, how long will it take for her to have enough money for her project?
Answer:
[tex]t=\frac{ln[\frac{A}{x}]}{n*ln[1+\frac{R}{n}]}[/tex]Step-by-step explanation:
As the question is not complete, we will generalize the statement as follows:
Olanda needs "A" for a future project. She can invest "x" now at an annual rate of "R" , compounded monthly. Assuming that no withdrawals are made, how long will it take for her to have enough money for her project?
A= Amount Olanda needs (future value).
X= Amount available for Olanda.
R= Anual rate.
First, we have that the future value is given by:
[tex]A=x(1+\frac{R}{n}) ^{nt}[/tex]
Where n=12 because the rate is annual.
And we need to calculate the time it will take with that annual rate to get the money she needs (future value) from what she has now (present value). So, we must make it explicit for "t"; solving:
We have:
[tex]\frac{A}{x} = (1+ \frac{R}{n})^{nt} [/tex]
[tex]ln[\frac{A}{x}]=ln[ (1+ \frac{R}{n})^{nt}][/tex]
Applying logarithmic properties
[tex]ln[\frac{A}{x}]=nt*ln[1+\frac{R}{n}][/tex]
Finally, we have:
[tex]t=\frac{ln[\frac{A}{x}]}{n*ln[1+\frac{R}{n}]}[/tex]A manufacturer of coffee vending machines has designed a new, less expensive machine. The current machine is known to dispense (into cups) an average of 7 fl. oz., with a standard deviation of .2 fl. oz. When the new machine is tested using 15 cups, the mean and the standard deviation of the fills are found to be 7 fl. oz. and .219 fl. oz. Test H0: σ = .2 versus Ha: σ ≠ .2 at levels of significance .05 and .01. Assume normality. (Round your answer to 4 decimal places.)
Answer:
[tex] t=(15-1) [\frac{0.219}{0.2}]^2 =16.7864[/tex]
[tex]p_v = 2*P(\chi^2_{14}>16.786)=0.5355[/tex]
And the 2 is because we are conducting a bilateral test.
[tex]\alpha=0.05[/tex] since the the [tex]p_v >0.05[/tex] we fail to reject the null hypothesis.
[tex]\alpha=0.01[/tex] since the the [tex]p_v >0.01[/tex] we fail to reject the null hypothesis.
Step-by-step explanation:
Data given
[tex]\mu=7[/tex] population mean (variable of interest)
[tex]\sigma=0.2[/tex] represent the population standard deviation
[tex]s=0.219[/tex] represent the sample deviation
n=15 represent the sample size
[tex]\alpha=0.05,0.01[/tex] represent the values for the significance level
Hypothesis test
On this case we want to check if the population standard deviation is equal or not to 0.2, so the system of hypothesis are:
H0: [tex]\sigma = 0.2[/tex]
H1: [tex]\sigma \neq 0.2[/tex]
In order to check the hypothesis we need to calculate the statistic given by the following formula:
[tex] t=(n-1) [\frac{s}{\sigma_o}]^2 [/tex]
This statistic have a Chi Square distribution distribution with n-1 degrees of freedom.
What is the value of your test statistic?
Now we have everything to replace into the formula for the statistic and we got:
[tex] t=(15-1) [\frac{0.219}{0.2}]^2 =16.7864[/tex]
P value
We know the degrees of freedom of the distribution 14 on this case and we can find the p value like this:
[tex]p_v = 2*P(\chi^2_{14}>16.786)=0.5355[/tex]
And the 2 is because we are conducting a bilateral test.
[tex]\alpha=0.05[/tex] since the the [tex]p_v >0.05[/tex] we fail to reject the null hypothesis.
[tex]\alpha=0.01[/tex] since the the [tex]p_v >0.01[/tex] we fail to reject the null hypothesis.
The question involves hypothesis testing for variance in relation to a coffee vending machine. After calculating the test statistic (using the Chi-square test) and comparing it with critical values at .05 and .01 significance levels, we fail to reject the null hypothesis. This means that there's not enough statistical evidence to suggest that the new machine has different variability in coffee amounts dispensed compared to the old machine.
Explanation:To solve this problem, you need to use hypothesis testing for variance. Hypothesis testing is a statistical method that is used in making statistical decisions using experimental data. In hypothesis testing, an initial claim or belief about a population is translated into two competing hypotheses, the null hypothesis and the alternative hypothesis.
In this case, the null hypothesis(H0) is that the variance σ² is equal to 0.04 (σ=0.2; σ^2 = (0.2)² = 0.04), and the alternative hypothesis(Ha) is that the variance is not equal to .04.
The test statistic for this hypothesis is the Chi-Square statistic. It's calculated using the formula:
[ (n-1)*s² ] / σ²
where n is the sample size, σ² is the variance under the null hypothesis and s² is the empirical or sample variance. In your case, n = 15, σ² = 0.04 and s² = (0.219)² = 0.047961.
Substituting the values, we get (chi-square) χ² = (15-1)*(0.047961)/0.04 = 0.5995125.
The degrees of freedom (df) in this context is (n-1) = 14. At .05 level of significance, for 14 degrees of freedom, the critical values of χ² are 6.571 and 23.684. Likewise, for .01 level of significance, critical values are 3.787 and 30.578.
The calculated statistic (0.5995125) falls between these ranges for both cases, so we fail to reject the null hypothesis at both .05 and .01 significance level. Hence, the manufacturer does not have enough evidence to suggest that the new machine has a different variability in the amount of coffee dispensed compared to the old machine.
Learn more about Hypothesis Testing here:https://brainly.com/question/34171008
#SPJ12
Choose the property used to rewrite the expression. log618-log66 = log63.a) Commutative Propertyb) Product Propertyc) Power Propertyd) Quotient Property
Answer:
Option d) is correct
ie, quotient property
Step-by-step explanation:
Given expression is log 618-log 66=log 63
Now we take log 618-log 66
log 618-log 66 = log [tex]\frac{618}{66}[/tex] [By using quotient property, log[tex]\frac{x}{y}=\log x- \log y[/tex]]
= log 63
Therefore log 618- log 66= log 63
Option d) is correct
In the given expression we are using the quotient property
A paper describes a study in which researchers observed wait times in coffee shops in Boston. Both wait time and gender of the customer were observed. The mean wait time for a sample of 145 male customers was 85.8 seconds. The mean wait time for a sample of 141 female customers was 113.4 seconds. The sample standard deviations (estimated from graphs that appeared in the paper) were 50 seconds for the sample of males and 75 seconds for the sample of females. For purposes of this exercise, suppose that these two samples are representative of the populations of wait times for female coffee shop customers and for male coffee shop customers. Is there convincing evidence that the mean wait time differs for males and females? Test the relevant hypotheses using a significance level of 0.05. (Use a statistical computer package to calculate the P-value. Use μmales − μfemales. Round your test statistic to two decimal places, your df down to the nearest whole number, and your P-value to three decimal places.)
Answer:
[tex]t=\frac{85.8-113.4}{\sqrt{\frac{(50)^2}{145}+\frac{(75)^2}{141}}}}=-3.65[/tex]
[tex]p_v =2*P(t_{(284)}<-3.65)=0.000312[/tex]
Using Excel we can use the following code: "=2*T.DIST(-3.65;284;TRUE)"
So the p value is a very low value and using any significance level giveen [tex]\alpha=0.05[/tex] always [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and there is enough evidence to conclude that the means are different.
Step-by-step explanation:
1) Data given and notation
[tex]\bar X_{M}=85.8[/tex] represent the mean for the males
[tex]\bar X_{F}=113.4[/tex] represent the mean for females
[tex]s_{M}=50[/tex] represent the sample standard deviation for males
[tex]s_{F}=75[/tex] represent the sample standard deviation for female
[tex]n_{M}=145[/tex] sample size for the group poisoned
[tex]n_{F}=141[/tex] sample size for the group unpoisoned
t would represent the statistic (variable of interest)
2) Concepts and formulas to use
We need to conduct a hypothesis in order to check if the mean for the two groups are equal or not , the system of hypothesis would be:
Null hypothesis:[tex]\mu_{M} = \mu_{F}[/tex]
Alternative hypothesis:[tex]\mu_{M} \neq \mu_{F}[/tex]
If we analyze the size for the samples both are higher than 30 but the population deviations are not given, so for this case is better apply a t test to compare means, and the statistic is given by:
[tex]t=\frac{\bar X_{M}-\bar X_{F}}{\sqrt{\frac{s^2_{M}}{n_{M}}+\frac{s^2_{F}}{n_{F}}}}[/tex] (1)
t-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.
3) Calculate the statistic
We can replace in formula (1) the results obtained like this:
[tex]t=\frac{85.8-113.4}{\sqrt{\frac{(50)^2}{145}+\frac{(75)^2}{141}}}}=-3.65[/tex]
4) Statistical decision
The first step is calculate the degrees of freedom, on this case:
[tex]df=n_{M}+n_{F}-2=145+141-2=284[/tex]
Since is a two tailed test the p value would be:
[tex]p_v =2*P(t_{(284)}<-3.65)=0.000312[/tex]
Using Excel we can use the following code: "=2*T.DIST(-3.65;284;TRUE)"
So the p value is a very low value and using any significance level giveen [tex]\alpha=0.05[/tex] always [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and there is enough evidence to conclude that the means are different.
Part-time weekly earnings ($) by college students.
hours worked (x) weekly pay (y)
10 93
15 171
20 204
20 156
35 261
this is the question: a) make an excel scatter plot. what does it suggest about the population correlation between X and Y? b) make an excel worksheet to calculate SSxx,SSyy, and SSxy c) use appendix D to find t.05 for a two-tailored test for zero correlation. d) calculate the t test statistics. can you reject p=0?
Answer:
There is a significant linear correlation between the two variables.
Step-by-step explanation:
Given that part time weekly earnings are
x y
10 93
15 171
20 204
20 156
35 261
Correlation 0.9199
H0: r=correlatin =0
Ha: r ≠0
(two tailed test)
r difference = 0.9199
t = [tex]r*\sqrt{(n-2)/(1-r^2)} \\=4.063[/tex]
p value < 0.0001
Since p <0.05 we reject null hypothesis
There is a significant linear correlation between the two variables.
This answer demonstrates how to create a scatter plot in Excel, interpret the correlation, calculate SSxx, SSyy, and SSxy, find t.05 from a t-distribution table, and perform a t-test to determine if the correlation coefficient is significantly different from zero.
Explanation:In Excel, you can create a scatter plot by selecting the appropriate data and choosing Scatter Plot under the Insert tab. The scatter plot suggests a positive correlation between hours worked (X) and weekly pay (Y), indicating that as hours worked increase, so does weekly pay.
To calculate SSxx, SSyy, and SSxy, first, calculate the average of X and Y. Next, subtract the average from each individual data point and square the result to find SSxx and SSyy. For SSxy, subtract the average of X from each X data point and the average of Y from each Y data point, multiply these together, and take the sum.
With regards to finding t.05, look at the t-distribution table available with your study materials. Lastly, calculate the t test statistics using the formula t = r * sqrt((n-2)/(1-r^2)), where n is the number of pairs, r is the correlation coefficient obtained from your Excel scatter plot, and sqrt is the square root function. If the calculated t-value exceeds the t.05 value, you reject the null hypothesis (p = 0).
Learn more about Statistical Analysis here:https://brainly.com/question/33812621
#SPJ6
A culture of yeast grows at a rate proportional to its size. If the initial population is 1000 cells and it doubles after 4 hours, answer the following questions.1)Write an expression for the number of yeast cells after tt hours.Answer: P(t)=2) Find the number of yeast cells after 10 hours?3) Find the rate at which the population of yeast cells is increasing at 1010 hours.Answer (in cells per hour):??
1. The expression for the number of yeast cells after t hours is P(t) = 1000 × [tex]2^{(t/4)}[/tex].
2. After 10 hours, there will be 4000 yeast cells.
3. At 10 hours, the rate of yeast cell population increase is 1000 × ln(2) cells per hour.
1. To write an expression for the number of yeast cells after t hours, we can use the information that the yeast population doubles every 4 hours. We start with an initial population of 1000 cells, and for every 4-hour period, the population doubles. Therefore, the expression can be written as:
P(t) = 1000 ×[tex]2^{(t/4)}[/tex]
Where P(t) represents the number of yeast cells after t hours.
2. To find the number of yeast cells after 10 hours, we can simply plug t = 10 into the expression we derived:
P(10) = 1000 × [tex]2^{(10/4)}[/tex]
P(10) = 1000 × [tex]2^{(2)}[/tex]
P(10) = 1000 × 4
P(10) = 4000
So, there will be 4000 yeast cells after 10 hours.
3. To find the rate at which the population of yeast cells is increasing at 10 hours, we can take the derivative of the expression P(t) with respect to t and evaluate it at t = 10:
P'(t) = (1000/4) × [tex]2^{(t/4)}[/tex] × ln(2)
P'(10) = (1000/4) × [tex]2^{(10/4)}[/tex] × ln(2)
P'(10) = (1000/4) × [tex]2^{(2)}[/tex] × ln(2)
P'(10) = 250 × 4 ln(2)
P'(10) = 1000 × ln(2)
So, the rate at which the population of yeast cells is increasing at 10 hours is 1000× ln(2) cells per hour.
For more such questions on population
https://brainly.com/question/30396931
#SPJ3
Complete question below :
1. Write an expression for the number of yeast cells after t hours.
2. Find the number of yeast cells after 10 hours.
3. Find the rate at which the population of yeast cells is increasing at 10 hours.
An exponential growth model for yeast growth gives P(t)=1000e^(ln(2)t/4). After 10 hours, the yeast population is approximately 5657 cells. The rate of increase at 10 hours is about 978 cells per hour.
Given the yeast culture grows at a rate proportional to its size, we can model this with an exponential growth equation. The general form is:
P(t) = P0ekt
where P(t) is the population at time t, P0 is the initial population, k is the growth constant, and t is the time in hours.
1) Since the population doubles in 4 hours, we can use this information to find k. Start with the equation when the population doubles:
P(4) = 2P0
Substitute P0 and t = 4:
2P0 = P0e4k
Divide both sides by P0:
2 = e4k
Take the natural logarithm of both sides:
ln(2) = 4k
Solve for k:
k = ln(2) / 4
Now substitute k back into the general equation:
P(t) = 1000e(ln(2)t/4)
2) To find the number of yeast cells after 10 hours, substitute t = 10:
P(10) = 1000e(ln(2)10/4)
Simplify the exponent:
P(10) = 1000e(2.5ln(2))
P(10) = 1000 * 22.5
P(10) ≈ 1000 * 5.657
P(10) ≈ 5657 cells
3) To find the rate of increase at 10 hours, we need to differentiate P(t):
dP/dt = 1000 * (ln(2) / 4) * e(ln(2)t/4)
Substitute t = 10:
dP/dt = 1000 * (ln(2) / 4) * 2(10/4)
dP/dt = 1000 * (ln(2) / 4) * 2.5
dP/dt ≈ 1000 * 0.173 * 5.657
dP/dt ≈ 978 cells per hour
Complete question below :
A culture of yeast grows at a rate proportional to its size. If the initial population is 1000 cells and it doubles after 4 hours, answer the following questions:
1. Write an expression for the number of yeast cells after t hours.
2. Find the number of yeast cells after 10 hours.
3. Find the rate at which the population of yeast cells is increasing at 10 hours.
The human eye can detect amounts of light that differ by a factor of.
100
500
2,000
8
10,000
Answer:
10000
Step-by-step explanation:
A report summarizes a survey of people in two independent random samples. One sample consisted of 700 young adults (age 19 to 35) and the other sample consisted of 200 parents of children age 19 to 35. The young adults were presented with a variety of situations (such as getting married or buying a house) and were asked if they thought that their parents were likely to provide financial support in that situation. The parents of young adults were presented with the same situations and asked if they would be likely to provide financial support to their child in that situation. (a) When asked about getting married, 41% of the young adults said they thought parents would provide financial support and 43% of the parents said they would provide support. Carry out a hypothesis test to determine if there is convincing evidence that the proportion of young adults who think parents would provide financial support and the proportion of parents who say they would provide support are different. (Use α = 0.05. Use a statistical computer package to calculate the P-value. Use μyoung adults − μparents. Round your test statistic to two decimal places and your P-value to three decimal places.)
Answer:
[tex]z=\frac{0.41-0.43}{\sqrt{\frac{0.41(1-0.41)}{700}+\frac{0.43(1-0.43)}{200}}}=-0.51[/tex]
[tex]p_v =2*P(Z<-0.51)=0.614[/tex]
And we can use the following R code to find it: "2*pnorm(-0.505)"
The p value is a very high value and using any significance given [tex]\alpha=0.05[/tex] always [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the two proportions are not significantly different.
Step-by-step explanation:
1) Data given and notation
[tex]n_{1}=700[/tex] sample of young adults (age 19 to 35)
[tex]n_{2}=200[/tex] sample of children age 19 to 35
[tex]p_{1}=0.41[/tex] represent the proportion of young adults said they thought parents would provide financial support
[tex]p_{2}=0.43[/tex] represent the proportion of parents said they would provide support
[tex]\alpha=0.05[/tex] represent the significance level
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the value for the test (variable of interest)
2) Concepts and formulas to use
We need to conduct a hypothesis in order to check if the proportion for the two samples are different , the system of hypothesis would be:
Null hypothesis:[tex]p_{1} = p_{2}[/tex]
Alternative hypothesis:[tex]p_{1} \neq p_{2}[/tex]
We need to apply a z test to compare proportions, and the statistic is given by:
[tex]z=\frac{p_{1}-p_{2}}{\sqrt{\frac{p_1 (1-p_1)}{n_{1}}+\frac{p_2 (1-p_2)}{n_{2}}}}[/tex] (1)
3) Calculate the statistic
Replacing in formula (1) the values obtained we got this:
[tex]z=\frac{0.41-0.43}{\sqrt{\frac{0.41(1-0.41)}{700}+\frac{0.43(1-0.43)}{200}}}=-0.51[/tex]
4) Statistical decision
We can calculate the p value for this test.
Since is a two tailed test the p value would be:
[tex]p_v =2*P(Z<-0.51)=0.614[/tex]
And we can use the following R code to find it: "2*pnorm(-0.505)"
The p value is a very high value and using any significance given [tex]\alpha=0.05[/tex] always [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the two proportions are not significantly different.
Look at the steps used when solving 3(x - 2) = 3 for x Which step is the result of combining like terms?
A) Step 1
B) Step 2
C) Step 3
D) Step 4
Answer:
Step 1
Step-by-step explanation:
Like terms are mathematical terms that have the exact same variables and exponents, this is why step 1 is the answer.
Answer : The correct option is, (B) Step 2
Step-by-step explanation :
The given expression is:
3(x - 2) = 3
In this expression, 'x' is a variable.
By using distributive property, we get:
3x - 6 = 3
Now adding 6 on both side, we get:
3x - 6 + 6 = 3 + 6
Now combining like terms, we get:
3x = 9
Now dividing the expression by 3, we get the value of 'x'.
x = 3
The meaning of like terms in mathematics is that have the same variables and exponents.
Hence, the step result of combining like terms is, Step 2
A company that produces garden hoses claims their product has a lifespan of at least 20 years. Thus, they came up with the following hypothesis test.
H0 : μ ≥ 20
Ha : μ < 20
A sample of 50 garden hoses provided a mean lifespan of 19.4 years, with a (population) standard
deviation of 2.
(a) Compute the z-value test statistic
B) what is the p-value?
c) Using α = 0.05, do you Reject H0 or Fail to reject H0?
Answer:
(a) z-value = -2.12
(b) p-value = 0.0170
(c) Reject H0
Step-by-step explanation:
(a)
[tex]std-err=\frac{std-dev}{\sqrt{n}}=\frac{2}{\sqrt{50}}=0.2828[/tex]
[tex]z-value=\frac{X-mean}{std-err}=\frac{19.4-20}{0.2828}=-2.12[/tex]
(b)
with
z-value = -2.12
significance level = 0.05
one-tail hypothesis (H0: μ ≥ 20)
We can see on the normal distribution table (Z-Score table) that
p-value = 0.0170
(c)
Since p value (0.0170) is less than α=0.05 we reject H0
Hope this helps!
The Normal approximation (with continuity correction) of the probability P(13 < X ≤ 16) is equal to:_________
Answer:
The Normal approximation (with continuity correction) of the probability P(13 < X ≤ 16) is equal to: P(13.5<X<16.5).
Step-by-step explanation:
This question is intended to calculate the probability fo a variable that follows a binomial distribution to be between 13 and 16.
When approximated to a Normal distribution, a correction for continuity has to be made, because the binomial distribution is a discrete value function and the normal function is a continous value function.
For X>13, it should be rewritten as X>13.5 (it substracts because it does not include 13).
For X≤16, it should be rewritten as X<16.5 (it adds because it includes 16).
The Normal approximation (with continuity correction) of the probability P(13 < X ≤ 16) is equal to: P(13.5<X<16.5).
subtract: (−2x2 + 9x − 3) − (7x2 − 4x + 2) −9x2 − 13x + 5 −9x2 + 13x − 5 5x2 − 13x − 1 5x2 + 5x − 5
Answer:
17x² - 5x + 11 = 0
Step-by-step explanation:
To add or subtract a polynomial do the operation with like - terms.
i.e., Two terms of x² gets added or subtracted. Similarly terms of x and constant.
Here, we have to subtract:
[tex]$ 2x^2 + 9x - 3 - 7x^2 + 4x - 8 - 9x^2 - 13x + 5 - 9x^2 + 13x - 5 + 5x^2 - 13x - 1 + 5x^2 + 5x - 5 $[/tex]
Club all the like terms for easier simplification. We get:
[tex]$ (-2 - 7 - 9 - 9 + 5 + 5)x^2 + (9 + 4 - 13 + 13 - 13 + 5)x + (-3 - 2 + 5 - 5 - 1) $[/tex]
[tex]$ \implies - 17 x^2 + 5x - 11 = 0 $[/tex]
Multiplying by -1 throughout:
17x² - 5x + 11 = 0 is the answer.
The lifetime (in hours) of a 60-watt light bulb is a random variable that has a Normal distribution with σ = 30 hours. A random sample of 25 bulbs put on test produced a sample mean lifetime of = 1038. Construct a 92% confidence interval estimate for the mean lifetime μ. If it were desired to cut the confidence interval to half its length while keeping the same 92% level, what size sample would be required to achieve this?
Answer:
a) 92% Confidence interval: (1027.5,1048.5)
b) Sample size = 100
Step-by-step explanation:
We are given the following in the question:
Sample mean, [tex]\bar{x}[/tex] = 1038
Sample size, n = 25
Alpha, α = 0.08
Population standard deviation, σ = 30
a) 92% Confidence interval:
[tex]\mu \pm z_{critical}\frac{\sigma}{\sqrt{n}}[/tex]
Putting the values, we get,
[tex]z_{critical}\text{ at}~\alpha_{0.08} = \pm 1.75[/tex]
[tex]1038 \pm 1.75(\frac{30}{\sqrt{25}} ) = 1038 \pm 10.5 = (1027.5,1048.5)[/tex]
b) In order to reduce the confidence interval by half, we have to quadruple the sample size.
Thus,
[tex]\text{Sample size} = 25\times 4 = 100[/tex]
A 92% confidence interval for the mean lifetime of a 60-watt bulb is (1027.5, 1048.5) hours. To halve this interval while maintaining the same confidence level, the sample size would need to be increased to 100.
Explanation:The lifetime of a light bulb, expressed as a random variable, is said to have a Normal distribution with σ = 30 hours. The given random sample consists of 25 bulbs with a sample mean lifetime of = 1038. To construct a 92% confidence interval estimate for the mean lifetime (μ), we first need to identify the standard error, which is σ/√n => 30/√25 = 6. To calculate the confidence interval, we adjust the sample mean by a few standard errors. For a 92% confidence interval, the Z score is approximately ±1.75 (obtained from a Z distribution table). Therefore, the interval is 1038 ± 1.75 * 6 = 1038 ± 10.5. Hence, the 92% confidence interval for the population mean is (1027.5, 1048.5). To halve the confidence interval at the same confidence level (i.e. to make it ±5.25), we need to halve the standard error. Since the standard error is inversely proportional to the square root of the sample size, we need to quadruple the sample size to halve it. So, a sample size of 100 would be required.
Learn more about Confidence Interval here:https://brainly.com/question/34700241
#SPJ11
Using traditional methods, it takes 108 hours to receive a basic driving license. A new license training method using Computer Aided Instruction (CAI) has been proposed. A researcher used the technique with 270 students and observed that they had a mean of 107 hours. Assume the variance is known to be 49. A level of significance of 0.1 will be used to determine if the technique performs differently than the traditional method. Find the value of the test statistic. Round your answer to 2 decimal places. Enter the value of the test statistic.
Answer: z= -2.35
Step-by-step explanation:
When the population variation is know, then we calculate the z-statistic.
Formula to calculate the z-test statistic is given by :-
[tex]z=\dfrac{\overline{x}-\mu}{\dfrac{s}{\sqrt{n}}}[/tex]
where , n= sample size
[tex]\overline{x}[/tex] = sample mean
[tex]\mu[/tex] = Population mean
[tex]\sigma[/tex] =Population standard deviation.
As per given we have,
n= 270
[tex]\mu= 108[/tex]
[tex]\overline{x}=107[/tex]
[tex]\sigma^2=49\\\\\Rightarrow \sigma=\sqrt{49}=7[/tex]
Then, z-statistic will be
[tex]z=\dfrac{107-108}{\dfrac{7}{\sqrt{270}}}[/tex] (Substitute all the values )
[tex]\Rightarrow\ z=\dfrac{-1}{\dfrac{7}{16.4316767252}}[/tex]
[tex]\Rightarrow\ z=\dfrac{-1}{0.426006433614}[/tex]
[tex]\Rightarrow\ z=-2.34738238931\approx-2.35[/tex]
Hence, the value of the test statistic= z= -2.35