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Answer:

[tex]s_{p_1-p_2}=\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2} }[/tex]

Step-by-step explanation:

The formula for the standard error of the difference between the estimates od the population proportions is:

[tex]s_{p_1-p_2}=\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2} }[/tex]

This is expected, as the variance of a sum (or a substraction) of two random variables is equal to the sum of the variance of the two variables.

Then, the standard error (or standard deviation) is the square root of this variance.

Answer:

Critical value: [tex]z= 2.575[/tex]

99% confidence interval: (0.695 cc/cubic meter, 0.713 cc/cubic meter).

Step-by-step explanation:

We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:

[tex]\alpha = \frac{1-0.99}{2} = 0.005[/tex]

Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex].

So it is z with a pvalue of [tex]1-0.005 = 0.995[/tex], so [tex]z = 2.575[/tex] is the critical value

Now, find M as such

[tex]M = z*\frac{\sigma}{\sqrt{n}} = 2.575*\frac{0.0142}{\sqrt{17}} = 0.0089[/tex]

The lower end of the interval is the mean subtracted by M. So it is 0.704 - 0.0089 = 0.695 cc/cubic meter.

The upper end of the interval is the mean added to M. So it is 0.704 + 0.0089 = 0.713 cc/cubic meter.

So

99% confidence interval: (0.695 cc/cubic meter, 0.713 cc/cubic meter).

Answer:

a) Q₀ × 1.335 cps

b) 12 fourths

c)  5 octaves

Step-by-step explanation:

The circle of fourth is generated by starting at any note and stepping upward by intervals of a fourth(five half-steps)

(a)  By what factor is the frequency of a tone increased if it is raised by a​ fourth?

consider the following exponential growth formula :

Q = Q₀ × f °

Q = Q₀ × 1.05946 ⁿ

Substituting n = 5

Q = Q₀ × 1.05946 ⁿ

Q = Q₀  × 1.05946⁵

Q = Q₀  × 1.335 cps

Therefore, note that  five half-step higher will be increased by factor 1.335

(b)  How many fourths are required to complete the entire circle of​ fourths?

SOLUTION

In each step, number of half steps = 5

Total number of half steps in one octave = 12 half- steps

Therefore, total number of fourths required to complete the entire circle of fourths = 12 fourths

(c) Total number of half steps in one octave = 12 half-steps  

Total half steps in complete circle of fourths  

= 12×5  

= 60 half-steps

Calculating number of octaves (dividing it by 12)  

= 60/12  

= 5 octaves

It is given that the circle of fourth is generated by starting at any note and stepping upward by intervals of a fourth(five half-steps)

[tex]Q = Q_0 \times f ^0\\Q = Q_0 \times 1.05946^n[/tex]

Put n = 5 in the above formula, we get

[tex]Q = Q_0 ( 1.05946 ^n)\\Q = Q_0 ( 1.05946 ^5)\\Q = Q_0 (1.335) cps[/tex]

So, by 1.335  factor the frequency of a tone increased if it is raised by a​ fourth.

Total number of half steps in one octave = 12 half- steps

So, total number of fourths required to complete the entire circle of fourths = 12 fourths

Total half steps in complete circle of fourths will be  12×5  = 60 half-steps

Now, the number of octaves is [tex]\frac{60}{12} =5[/tex]  octaves.

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Answer:

There is no evidence that there is no significant difference between the sample means

Step-by-step explanation:

given that a  statistics instructor who teaches a lecture section of 160 students wants to determine whether students have more difficulty with one-tailed hypothesis tests or with two-tailed hypothesis tests. On the next exam, 80 of the students, chosen at random, get a version of the exam with a 10-point question that requires a one-tailed test. The other 80 students get a question that is identical except that it requires a two-tailed test. The one-tailed students average 7.81 points, and their standard deviation is 1.06 points

The two-tailed students average 7.64 points, and their standard deviation is 1.33 points.

Group   One tailed X     Two tailed Y  

Mean 7.8100 7.6400

SD 1.0600 1.3300

SEM 0.1185 0.1487

N 80       80  

[tex]H_0:\bar x=\bar y\\H_a: \bar x \neq \bar y[/tex]

(Two tailed test)

The mean of One tailed X minus Two tailed Y equals 0.1700

t = 0.8940

 df = 158

p value =0.3727

 p is greater than alpha 0.05

There is no evidence that there is no significant difference between the sample means

Answer:

Her usual driving speed is 38 miles per hour.

Step-by-step explanation:

We know that:

[tex]s = \frac{d}{t}[/tex]

In which s is the speed, in miles per hour, d is the distance, in miles, and t is the time, in hours.

We have that:

At speed s, she takes two hours to drive. So

[tex]s = \frac{d}{2}[/tex]

[tex]d = 2s[/tex]

However, on one particular trip, after 40% of the drive, she had to reduce her speed by 30 miles per hour, driving at this slower speed for the rest of the trip. This particular trip took her 228 minutes.

228 minutes is 3.8 hours. So

[tex]0.4s + 0.6(s - 30) = \frac{d}{3.8}[/tex]

So

[tex]3.8(0.4s + 0.6s - 18) = d[/tex]

[tex]3.8s - 68.4 = 2s[/tex]

[tex]1.8s = 68.4[/tex]

[tex]s = 38[/tex]

Her usual driving speed is 38 miles per hour.

Answer:

112

Step-by-step explanation:

Let A be a subset of S that satisfies such condition.

If 3∈A, then the other three elements of A must be chosen from the set B={1,2,5,6,7,8,9,10} (because 3 cannot be chosen again and 4 can't be alongside 3). B has eight elements, then there are [tex]\binom{8}{3}=56[/tex] ways to select the remaining elements of A (the binomial coefficient counts this). The remaining elements determine A uniquely, then there are 56 subsets A.

If 4∉A, we have to choose the remaining elements of A from the set B={1,2,5,6,7,8,9,10}. B has eight elements, then there are [tex]\binom{8}{3}=56[/tex] ways to select the remaining elements of A. Thus, there are 56 choices for A.

By the sum rule, the total number of subsets is 56+56=112

Answer:a is 40 degrees

b is 140 degrees

Step-by-step explanation:

The given polygon has 5 irregular sides. This means that it is an irregular pentagon. The sum of the exterior angles of a polygon is 360 degrees

The exterior angles of the given pentagon are a, 75, 65, 60 and 120. Therefore

a + 75 + 65 + 60 + 120 = 360

a + 320 = 360

Subtracting 320 from both sides of the equation, it becomes

a = 360 - 320 = 40 degrees

The sum of angles on a straight line is 180/degrees. Therefore,

a + b = 180

b = 180 - a = 180 - 40 = 140 degrees

Answer:

Step-by-step explanation:

Answer:

a=2

b=6

Step-by-step explanation:

Assuming a uniform distribution and that a ≤ x ≤ b, if f(x) = 1/4, then:

[tex]f(x) =\frac{1}{4}=\frac{1}{b-a}\\b-a = 4[/tex]

If P(X > 4) = 1/2, then:

[tex]\frac{1}{2} = 1-(\frac{x-a}{b-a})=1-(\frac{4-a}{4})\\a-4+4 =\frac{1}{2}*4\\ a=2\\b=4+a\\b=6[/tex]

The values for a and b are, respectively, 2 and 6.

Answer:

a) H0=Weight of detergent=500gr  (null hypotheses )

H1=Weight of detergent>500 gr. (alternative hypotheses )

b) X test.

Step-by-step explanation:

a)

The statement from which we are going to establish the assumptions is that supplied by the manufacturer, which says that the average weight of the detergent is 500 g. So:

H0=Weight of detergent=500gr  (null hypotheses )

H1=Weight of detergent>500 gr. (alternative hypotheses )

b)

You are not providing the source you mention where the sample of detergent boxes taken is located. So I will explain in general terms how the problem should be posed.

For this case, we can use Z test because the sample we are taking is larger than 30.

The first thing we have to do is calculate Z, which is given by:

[tex]Z=\frac{(X-Xav)}{SD}[/tex]

Where:

Xav=Average of x.

SD=Standard deviation.

Then, we look for the table and if the value we have just calculated is greater than that of the table, then we must reject the null hypothesis.

Answer:

D. All of the above

Step-by-step explanation:

By the given data, some persons were in the experiment and each of them was given with three meals, Low Calorie, Moderate Calorie and High Calorie. They results for the meal they liked is as below:

Low Calorie Meal got likes = 6

Moderate Calorie Meal got likes = 7

High Calorie Meal got likes = 17

so,

Option A is correct as High Calorie meal got 17 likes while Low Calorie meal got 6 likes.

Option B is also correct as Low Calorie meal got 6 likes while Moderate Calorie meal got 7 likes.

Option C is correct too as High Calorie meal got largest number of likes even more than the double of Low calorie and Moderate calorie meal so it was more than expected.

Final answer:

Based on the data provided, without performing an actual chi-squared test due to a lack of p-value or statistical data, the most supported conclusion is that participants liked the high calorie meal more than was expected (option C). This is inferred from the observed preference for the high calorie meal, which was chosen significantly more often (17 times) compared to its expected frequency (10 times).

Explanation:

The question presents a scenario where participants are asked to taste three types of meals with different calorie counts and choose the one they like best. The observed frequencies (fo) for each type of meal (low calorie, moderate calorie, and high calorie) are given as 6, 7, and 17, respectively. The expected frequencies (fe) are all 10, assuming no preference among the meals. Using a significance level of .05, we must employ a chi-squared test to determine if the observed preferences significantly differ from what was expected given no effect of calorie count.

The chi-squared test result for this would involve calculating the sum of squared differences between observed and expected frequencies, divided by the expected frequencies for each category and summing those values. However, since the data on the actual chi-squared statistic or the p-value are not provided, we cannot perform the exact calculation. Nonetheless, we can make a conclusion based on the observed and expected frequencies:

There is a notable preference for the high calorie meal, as it was chosen substantially more often than expected (17 vs. 10).

The low calorie meal was chosen less often than expected (6 vs. 10).

The moderate calorie meal was also chosen less than expected, but to a lesser extent (7 vs. 10).

Without a specific p-value or chi-squared statistic, we can't formally conclude that the participants liked the high calorie meal significantly more than the other meals, but the observed data suggest that might be the case. Also, we cannot definitively reject the possibility that the preference for the high-calorie meal is due to chance.

Therefore, the most reasonable conclusion based on the given information without formal test statistics would be option C: Participants liked the high calorie meal more than was expected.

This is in line with the general principle that a higher calorie meal tends to be preferred potentially due to higher fat content, which has been associated with better taste and satisfaction, particularly in fast-food meals. At a .05 level of significance and without the actual test statistic, option C is the most supported by the provided data.

Answer:

We can conclude that  the percentage of Americans that do not trust the media to report fully and accurately has increased since 1997 (P-value=0.00014).

Step-by-step explanation:

We have to perform an hypothesis test on a proportion.

The null and alternative hypothesis are:

[tex]H_0: \pi=0.46\\\\H_1: \pi>0.46[/tex]

The significance level is α=0.05.

The standard deviation is estimated as:

[tex]\sigma=\sqrt{\frac{\pi(1-\pi)}{N} } =\sqrt{\frac{0.46(1-0.46)}{1010} }=0.0157[/tex]

The z value for this sample is

[tex]z=\frac{p-\pi-0.5/N}{\sigma} =\frac{525/1010-0.46-0.5/1010}{0.0157} =\frac{0.52-0.46-0.00}{0.0157}=\frac{0.06}{0.0157} =3.822[/tex]

The P-value for z=3.822 is P=0.00014.

The P-value is smaller than the significance level, so the effect is significant. The null hypothesis is rejected.

We can conclude that  the percentage of Americans that do not trust the media to report fully and accurately has increased since 1997.

Answer:

We conclude that there is no change in the average weight since the implementation of a new breakfast and lunch program at the school.

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = 100 pounds

Sample mean, [tex]\bar{x}[/tex] = 98 pounds

Sample size, n = 35

Alpha, α = 0.05

Population standard deviation, σ = 7.5 pounds

First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 100\text{ pounds}\\H_A: \mu < 100\text{ pounds}[/tex]

We use one-tailed(left) z test to perform this hypothesis.

Formula:

[tex]z_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} }[/tex]

Putting all the values, we have

[tex]z_{stat} = \displaystyle\frac{98 - 100}{\frac{7.5}{\sqrt{35}} } = -1.577[/tex]

Now, [tex]z_{critical} \text{ at 0.05 level of significance } = -1.64[/tex]

We calculate the p-value with the help of standard normal table.

P-value = 0.057398

Since the p-value is greater than the significance level, we fail to reject the null hypothesis and accept is.

We conclude that there is no change in the average weight since the implementation of a new breakfast and lunch program at the school.

To calculate the p-value for the hypothesis test, we calculate the t-score using the given values and the formula. Then, we use the t-distribution table or a calculator to find the p-value associated with the t-score and degrees of freedom. The calculated p-value is approximately 0.1093.

In this problem, we are given the mean and standard deviation of a normal distribution for the weights of seventh-graders. The researcher believes that the new breakfast and lunch program has decreased the average weight. We are asked to calculate the p-value for an appropriate hypothesis test.

To calculate the p-value, we can use the t-distribution since the population standard deviation is unknown. We can calculate the t-score using the formula:

t = (sample mean - population mean) / (sample standard deviation / √sample size)

Plugging in the given values, we get t = (98 - 100) / (7.5 / √35) = -1.654. The degrees of freedom for this test is (sample size - 1) = (35 - 1) = 34.

Using the t-distribution table or a calculator, we can find the p-value associated with a t-score of -1.654 and 34 degrees of freedom. The p-value is approximately 0.1093.

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Answer:

A region R is revolved about the y-axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to y or using the shell method and integrating with respect to x.

Step-by-step explanation:

We assume this question :

Fill in the blanks: A region R is revolved about the y-axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to _ or using the shell method and integrating with respect to _____.

We can calculate the volume of a region when revolved about y-axis with two common methods: washer method and shell method. We need to take in count, that the general formula for these methods are different respect to the variable used to calculate the volume.

Since the region R is revolved about y-axis, disk/washer method needs to calculate the integral respect to y, and by the other hand the shell method will calculate the integral respect to x.

Washer method

[tex]V\approx \sum_{k=1}^n \pi r^2 h = \sum_{k=1}^n \pi f(y_k)^2 dy[/tex]

[tex]V= \pi \int_{c}^d f(y)^2 dy[/tex]

Shell method

[tex]V = \lim_{n\to\infty} 2\pi x_i f(x_i)\Delta x =\int_{a}^b 2\pi x f(x) dx[/tex]

So then the correct answer would be:

A region R is revolved about the y-axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to y or using the shell method and integrating with respect to x.

Answer:

a. gender (two levels: male, female)

d. class size (three levels: small, medium, large)

Step-by-step explanation:

The study is going to be made to identify how male and female students react to the class size(small, medium, large). So there are two correct options, since the gender and the class size are factors in this research.

The correct options are:

a. gender (two levels: male, female)

d. class size (three levels: small, medium, large)

In this example, the factors and levels are gender, academic performance, and class size.

The factors and levels in this example are:

a. Gender (two levels: male, female)

b. Academic performance (three levels: above average, average, below average)

c. Class size (two levels: small, large)

Therefore, the answer is a, b, and c.

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Answer:

Null hypothesis:[tex]\mu \geq 4.0[/tex]    

Alternative hypothesis:[tex]\mu < 4.00[/tex]    

[tex]t=\frac{3.48-4.00}{\frac{1.150075}{\sqrt{45}}}=-3.033077[/tex]    

[tex]df=n-1=45-1=44[/tex]

Since is a left-sided test the p value would be:    

[tex]p_v =P(t_{(44)}<-3.033077)=0.002025[/tex]    

If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the mean is significantly less than 4.00 at 5% of significance.  

Step-by-step explanation:

Data given and notation    

[tex]\bar X=3.48[/tex] represent the sample mean

[tex]s=1.150075[/tex] represent the sample standard deviation    

[tex]n=45[/tex] sample size    

[tex]\mu_o =4.00[/tex] represent the value that we want to test  

[tex]\alpha=0.05[/tex] represent the significance level for the hypothesis test.    

z would represent the statistic (variable of interest)    

[tex]p_v[/tex] represent the p value for the test (variable of interest)    

State the null and alternative hypotheses.    

We need to conduct a hypothesis in order to check if the mean is less than 4.00 :    

Null hypothesis:[tex]\mu \geq 4.0[/tex]    

Alternative hypothesis:[tex]\mu < 4.00[/tex]    

Since we don't know the population deviation, is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:    

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)    

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".    

Calculate the statistic    

We can replace in formula (1) the info given like this:    

[tex]t=\frac{3.48-4.00}{\frac{1.150075}{\sqrt{45}}}=-3.033077[/tex]    

P-value    

First we need to calculate the degrees of freedom given by:

[tex]df=n-1=45-1=44[/tex]

Since is a left-sided test the p value would be:    

[tex]p_v =P(t_{(44)}<-3.033077)=0.002025[/tex]    

Conclusion    

If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the mean is significantly less than 4.00 at 5% of significance.    

Answer:

The correct option is e) [tex]H_0: \mu_1=\mu_2\ and\ H_a:\mu_1\neq \mu_2[/tex]

Step-by-step explanation:

Consider the provided information.

Ten college students reported that they consume coffee but do not smoke cigarettes; these comprise the Coffee-Only group, or Group 1. Another ten college students reported that they consumed coffee and also smoked cigarettes; these comprise the Coffee + Cigarettes group, or Group 2.

The null hypothesis tells the population parameter is equal to the claimed value.

If there is no statistical significance in the test then it is know as the null which is denoted by [tex]H_0[/tex], otherwise it is known as alternative hypothesis which denoted by [tex]H_a[/tex].

The amount of coffee that  "coffee drinkers" consume on a weekly basis differs depending on whether the coffee drinker also smokes cigarettes.

Thus, the required hypothesis are: [tex]H_0: \mu_1=\mu_2\ and\ H_a:\mu_1\neq \mu_2[/tex]

Therefore, the correct option is e) [tex]H_0: \mu_1=\mu_2\ and\ H_a:\mu_1\neq \mu_2[/tex]

Answer:

a) [tex]P(X<2.52)=P(Z<\frac{2.52-2.55}{0.035})=P(Z<-0.857)=0.196[/tex]

b) [tex]P(\bar X <2.52) = P(Z<-1.714)=0.043[/tex]

c) [tex]P(\bar X <2.52) = P(Z<-4.286)=0.000[/tex]

d) For part a we are just finding the probability that an individual bottle would have a value of 2.52 liters or less. So we can't compare the result of part a with the results for parts b and c.

If we see part b and c are similar but the difference it's on the sample size for part b we just have a sample size 4 and for part c we have a sample size of 25. The differences are because we have a higher standard error for part b compared to part c.

Step-by-step explanation:

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".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  The letter [tex]\phi(b)[/tex] is used to denote the cumulative area for a b quantile on the normal standard distribution, or in other words: [tex]\phi(b)=P(z<b)[/tex]

Let X the random variable that represent the amount of water in a bottle of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(2.55,0.035)[/tex]

a. What is the probability that an individual bottle contains less than 2.52 liters?

We are interested on this probability

[tex]P(X<2.52)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

If we apply this formula to our probability we got this:

[tex]P(X<2.52)=P(\frac{X-\mu}{\sigma}<\frac{2.52-\mu}{\sigma})[/tex]

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

[tex]P(X<2.52)=P(Z<\frac{2.52-2.55}{0.035})=P(Z<-0.857)=0.196[/tex]

b. If a sample of 4 bottles is selected, what is the probability that the sample mean amount contained is less than 2.52 liters? (Round to three decimal places as noeded)

And let [tex]\bar X[/tex] represent the sample mean, the distribution for the sample mean is given by:

[tex]\bar X \sim N(\mu,\frac{\sigma}{\sqrt{n}})[/tex]

On this case  [tex]\bar X \sim N(2.55,\frac{0.035}{\sqrt{4}})[/tex]

The z score on this case is given by this formula:

[tex]z=\frac{\bar x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

And if we replace the values that we have we got:

[tex]z=\frac{2.52-2.55}{\frac{0.035}{\sqrt{4}}}=-1.714[/tex]

For this case we can use a table or excel to find the probability required:

[tex]P(\bar X <2.52) = P(Z<-1.714)=0.043[/tex]

c. If a sample of 25 bottles is selected, what is the probability that the sample mean amount contained is less than 2.52 liters? (Round to three decimal places as needed.)

The z score on this case is given by this formula:

[tex]z=\frac{\bar x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

And if we replace the values that we have we got:

[tex]z=\frac{2.52-2.55}{\frac{0.035}{\sqrt{25}}}=-4.286[/tex]

For this case we can use a table or excel to find the probability required:

[tex]P(\bar X <2.52) = P(Z<-4.286)=0.0000091[/tex]

d. Explain the difference in the results of (a) and (c)

For part a we are just finding the probability that an individual bottle would have a value of 2.52 liters or less. So we can't compare the result of part a with the results for parts b and c.

If we see part b and c are similar but the difference it's on the sample size for part b we just have a sample size 4 and for part c we have a sample size of 25. The differences are because we have a higher standard error for part b compared to part c.  

The problem involves using z-scores, normal distribution, central limit theorem and law of large numbers for probability calculations related to the amount of water in sampled bottles. The increase in sample size from 1 to 25 should show a higher probability for the sample's mean to be close to the population mean.

In this problem, we will apply the concept of the normal distribution and the central limit theorem to calculate probabilities and compare results. First, we will calculate the z-scores for (a), (b), and (c). The z-score formula is Z = (X - μ) / σ. In (a), X is 2.52 liters, μ (mean) is 2.55 liters, and σ (standard deviation) is 0.035 liters.

In (b) and (c), the σ of a sample mean is σ/√n, where n is the sample size (4 in (b) and 25 in (c)).

Next, we look up the z-score in the z-score table (or use a normal distribution calculator) to get the probabilities.

To answer part (d), probabilistically, as sample size increases, the sample mean tends to get closer to the population mean according to the law of large numbers. Hence, the probabilities in (c) should be higher than (a).

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M^2 + MNP

(3)^2 + (3)(4)(7)

9 + 12(7)

9 + 84

93

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Answer:

The equation would be [tex]30=12+9x[/tex].

Allen will Run for 2 hours.

Step-by-step explanation:

Given:

Total Distance traveled = 30 miles

Average rate of walking = 3 mi/hr

Average rate of running = 9 mi/hr

Time for walking = 4 hours

W need to find the time for running.

Solution:

Let Number of hrs required for running be 'x'.

Total Distance traveled is equal Distance Traveled in walking plus Distance traveled in Running

But Distance is equal Rate multiplied by Time.

Framing in equation form we get;

Hence Total Distance Traveled = Rate of walking × Hours of walking + Rate of Running × Hours of running.

Substituting the values we get;

[tex]30 = 3\times4 +9x\\\\30=12+9x[/tex]

Hence The equation would be [tex]30=12+9x[/tex]

Solving above equation we get;

[tex]9x=30-12\\\\9x=18\\\\x=\frac{18}{9}=2\ hrs[/tex]

Hence Allen will Run for 2 hours.

15x+10y≤270 and x+y≥20 are the constraints for this situation.

Step-by-step explanation:

Given,

Cost of long sleeved shirt = $15

Cost of short sleeved shirt = $10

Amount to spend = $270

Shirts to order = 20

Let,

Long sleeved shirts = x

Short sleeved shirts = y

At most means he can not spend more than $270, therefore,

15x+10y≤270

At least 20 means, he needs minimum 20 or more, therefore,

x+y≥20

15x+10y≤270 and x+y≥20 are the constraints for this situation.

Keywords: inequality, addition

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The value of the function g(x) at x = 2.9 and x = 3.1 will be -6.57 and -3.37, respectively. And the estimation is too small in part (a).

Integration is a way of finding the total by adding or summing the components. It's a reversal of differentiation, in which we break down functions into pieces. This approach is used to calculate the total on a large scale.

Suppose that we don't have a formula for g(x) but we know that g(3) = −5 and g'(x) = x2 + 7 for all x.

Integrate the function, then we have

∫g'(x) = ∫(x² + 7) dx

g(x) = x³/3 + 7x + c

g(3) = 3³ / 3 + 7 (3) + c

- 5 = 9 + 21 + c

c = - 30 - 5

c = -35

Then the function is given as,

g(x) = x³/3 + 7x - 35

At x = 2.9, we have

g(2.9) = (2.9)³/3 + 7(2.9) - 35

g(2.9) = -6.57

At x = 3.1, we have

g(3.1) = (3.1)³/3 + 7(3.1) - 35

g(3.1) = -3.37

The value of the function g(x) at x = 2.9 and x = 3.1 will be -6.57 and -3.37, respectively. And the estimation is too small in part (a).

More about the integration link is given below.

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Answer:

1) [tex]m=\frac{77}{10.5}=7.333[/tex]

2) [tex]b=\bar y -m \bar x=80.857-(7.333*3.5)=55.190[/tex]

3) [tex]\hat y=7.333(0)+55.190=55.190[/tex]

4) False. The values predited will fall on the same line since we are estimating  the values with just one line.

5) [tex]\hat y=7.333(4.5)+55.190=88.190[/tex]

6) [tex]R^2 = (0.971^2) =0.943[/tex]

And that means that the linear model explains 94.29% of the variation.

Step-by-step explanation:

We assume that the data is this one:

x:1.5,2.5, 3, 3.5 , 4, 4.5 ,5.5

y: 67, 69, 80, 81, 86, 89, 94.

Step 1 of 6: Find the estimated slope. Round your answer to three decimal places.

For this case we need to calculate the slope with the following formula:

[tex]m=\frac{S_{xy}}{S_{xx}}[/tex]

Where:

[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}[/tex]

[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}[/tex]

So we can find the sums like this:

[tex]\sum_{i=1}^n x_i = 1.5+2.5+3+3.5+4+4.5+5.5=24.5[/tex]

[tex]\sum_{i=1}^n y_i =67+ 69+ 80+ 81+ 86+ 89+ 94=566[/tex]

[tex]\sum_{i=1}^n x^2_i =1.5^2+2.5^2+3^2+3.5^2+4^2+4.5^2+5.5^2=96.25[/tex]

[tex]\sum_{i=1}^n y^2_i =67^2+69^2+80^2+81^2+86^2+89^2+94^2=46364[/tex]

[tex]\sum_{i=1}^n x_i y_i =1.5*67+2.5*69+3*80+3.5*81+4*86+4.5*89+5.5*94=2058[/tex]

With these we can find the sums:

[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=96.25-\frac{24.5^2}{7}=10.5[/tex]

[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=2058-\frac{24.5*566}{7}=77[/tex]

And the slope would be:

[tex]m=\frac{77}{10.5}=7.333[/tex]

Step 2 of 6: Find the estimated y-intercept. Round your answer to three decimal places.

Now we can find the means for x and y like this:

[tex]\bar x= \frac{\sum x_i}{n}=\frac{24.5}{7}=3.5[/tex]

[tex]\bar y= \frac{\sum y_i}{n}=\frac{566}{7}=80.857[/tex]

And we can find the intercept using this:

[tex]b=\bar y -m \bar x=80.857-(7.333*3.5)=55.190[/tex]

So the line would be given by:

[tex]\hat y=7.333 x +55.190[/tex]

Step 3 of 6: Determine the value of the dependent variable yˆ at x = 0.

[tex]\hat y=7.333(0)+55.190=55.190[/tex]

Step 4 of 6: Determine if the statement "Not all points predicted by the linear model fall on the same line" is true or false.

False. The values predited will fall on the same line since we are estimating  the values with just one line.

Step 5 of 6: Find the estimated value of y when x = 4.5. Round your answer to three decimal places.

[tex]\hat y=7.333(4.5)+55.190=88.190[/tex]

Step 6 of 6: Find the value of the coefficient of determination. Round your answer to three decimal places

n=7 [tex] \sum x = 24.5, \sum y = 566, \sum xy =2058, \sum x^2 =96.25, \sum y^2 =46364[/tex]  

And in order to calculate the correlation coefficient we can use this formula:

[tex]r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}[/tex]

[tex]r=\frac{7(2058)-(24.5)(566)}{\sqrt{[7(96.25) -(24.5)^2][7(46364) -(566)^2]}}=0.971[/tex]

And the determination coeffcient is just the square of the correlation coefficient given by:

[tex]R^2 = (0.971^2) =0.943[/tex]

And that means that the linear model explains 94.3% of the variation.

Regression analysis can be done to find the relationship between hours spent studying and midterm exam grades using the regression line equation yˆ = b0 + b1x. But doing this requires statistical computations and ensuring the correlation coefficient is statistically significant. The coefficient of determination, which indicates the proportion of the variance in the dependent variable predictable from the independent variable, is also needed.

The equation of the regression line you're referring to is written as yˆ = b0 + b1x, where yˆ is the predicted value of the dependent variable (in this case, Midterm Grades), b0 is the y-intercept, b1 is the slope or regression coefficient, and x is the value of the independent variable (Hours Studying). We can use the given data to find the estimated slope and y-intercept, value of yˆ at x = 0, test the truthfulness of the statement, and find the estimated value of y when x = 4.5.

However, being able to do this requires the use of statistical software or methods to calculate corresponding values. Also, this also requires us to know the value of the correlation coefficient to determine if it's statistically significant, as it's not wise to make predictions using a regression line with a correlation coefficient that's not statistically significant.

The last piece to find is the value of the coefficient of determination, which shows the proportion of the dependent variable's variance that's predictable from the independent variable(s). This value also needs computation using formulas or statistical software.

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Answer:

V = π (rh² − ⅓h³)

Step-by-step explanation:

Draw a cross section of the bowl.  Cut a thin, horizontal slice of the water.  This slice is a circular disc of radius x and thickness dy.  It is position a distance of y from the bottom of the bowl.  The volume of this slice is:

dV = πx² dy

By drawing a right triangle, we can define x in terms of y:

x² + (r−y)² = r²

x² + r² − 2ry + y² = r²

x² = 2ry − y²

Substitute:

dV = π (2ry − y²) dy

The total volume of the water is the sum of all the slices from y=0 to y=h.

V = ∫₀ʰ π (2ry − y²) dy

V = π ∫₀ʰ (2ry − y²) dy

V = π (ry² − ⅓y³) |₀ʰ

V = π (rh² − ⅓h³)

Final answer:

The formula to measure the volume of water in a hemispherical bowl is V = (2/3)πr²h.

Explanation:

The formula to measure the volume of water in a hemispherical bowl is V = (2/3)πr2h, where V is the volume, π is a constant approximately equal to 3.14159, r is the radius of the bowl, and h is the depth of the water.

1. The volume of a half of the globe is given by V = 2/3 * π * r³. However, since we are only interested in measuring the water's volume—which represents a portion of the hemisphere—we will need to modify the formula accordingly.

2. Within the larger hemisphere, a smaller one is created by the water's depth, h. To ascertain the volume of this more modest half of the globe, we can take away it from the volume of the bigger side of the equator.

3. The span of the more modest half of the globe is likewise r, since it has a similar ebb and flow as the bigger side of the equator.

4. The level of the more modest side of the equator, h, is equivalent to the profundity of the water.

5. To find the volume of the more modest side of the equator, we utilize the recipe V = 2/3 * π * r³ and substitute the sweep r and level h.

6. The volume of the water is around 50% of the volume of the more modest half of the globe since it just fills one side of the bowl.

7. In this way, the volume of the water in the hemispherical bowl is given by 1/2 * (2/3 * π * r³) = 1/3 * π * r³.

8. To represent the profundity of the water, we increase the volume by the proportion of the level h to the sweep r, giving us 1/3 * π * r² * h.

9. We can further simplify by rewriting the formula as 1/2 *  * h2 * (3r - h).

Answer:

option (d) 800,000

Step-by-step explanation:

Let the number of pairs that must be sold be 'x'

Thus,

Total variable cost for x units = $5x

Therefore,

Total cost = Fixed cost + Total variable cost

= $6 million + $5x

Now,

Profit = Cost - Revenue

Thus,

$2 million = $15x - ( $6 million + $5x )

or

$2 million + $6 million = $10x

or

$8 million = $10x

or

$8,000,000 = $10x

or

x = 800,000

Hence,

option (d) 800,000

Answer:

Probability = 0.100.

Step-by-step explanation:

For each adult, there are only two possible outcomes. Eithey they have one or more of these conditions. Or they do not. This means that the binomial probability distribution is used to solve this problem.

However, we are working with samples that are considerably big. So i am going to aproximate this binomial distribution to the normal.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

[tex]E(X) = np[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].

In this problem, we have that:

[tex]n = 750, p = 0.45[/tex]

So

[tex]E(X) = 750*0.45 = 337.5[/tex]

[tex]\sqrt{Var(X)} = \sqrt{750*0.45*0.55} = 13.62[/tex]

Calculate the probability that fewer than 320 out of the n = 750 adults over 65 in the study suffer from one or more of the conditions under consideration.

This is the pvalue of Z when [tex]X = 320[/tex]. So:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{320 - 337.5}{13.62}[/tex]

[tex]Z = -1.28[/tex]

[tex]Z = -1.28[/tex] has a pvalue of 0.100.

So

Probability = 0.100.

Answer:

1.83%

Step-by-step explanation:

For the expected value of 4 cases per 1000 children. We can use a formula for Poisson distribution to calculate the probability that none of the 1000 students would contract the flu:

[tex]P(X = 0) = \frac{\lambda^ke^{-\lambda}}{k!}[/tex]

[tex]P(X = 0) = \frac{4^0 e^{-4}}{0!}[/tex]

[tex]P(X = 0) = \frac{1}{e^4} = 0.0183 [/tex]

So the probability of this to happens is 1.83%

The probability that none of the 1,000 children will contract the flu after getting vaccinated, given the average number of flu cases after vaccination is approximately 0.0183 or 1.83%, using the Poisson distribution.

In this question, we are required to use the Poisson distribution to find the probability that none of the 1,000 children will contract the flu this season, given that the average number of cases per 1,000 children after vaccination is 4 (λ = 4).

A Poisson probability can be calculated using the formula P(x; λ) = (e^-λ * λ^x) / x!, where x represents the actual number of successes that result from the experiment, e is the base of the natural logarithm approximated to 2.71828, and λ is the mean number of successes that occur in a specified region.

To find the probability that zero children contract the flu, we use x = 0 in the formula. The Poisson probability P(0; 4) = ((e^-4 * 4^0) / 0!) = e^-4 = 0.0183.

So, the probability that none of the 1,000 children will contract the flu after getting vaccinated is approximately 0.0183 or 1.83%.

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The price of one ticket is $ 62.2

Given that a performer expects to sell 5000 tickets for an upcoming event

They want to make a total of $ 311, 000 in sales from these tickets

To find: price of one ticket

Let us assume that all tickets have the same price

Let "a" be the price of one ticket

So the total sales price of $ 311, 000 is obtained from product of 5000 tickets and price of one ticket

[tex]\text {total sales price }=5000 \times \text { price of one ticket }[/tex]

[tex]311000 = 5000 \times a\\\\a = \frac{311000}{5000}\\\\a = 62.2[/tex]

Thus the price of one ticket is $ 62.2

Answer:

Because the​ P-value is _(0.02)  less than the significance level 0.05​, there is sufficient evidence to support the claim that there is a linear correlation between lemon imports and crash fatality rates for a significance level of α=0.05.

C. The results suggest that imported lemons cause car fatalities.

Step-by-step explanation:

Hello!

The study variables are:

X₁: Weight of imported lemons.

X₂: Car crash fatality rate.

The objective is to test if the imported lemons affect the occurrence of car fatalities. To do so you are asked to use a linear correlation test.

I've made a Scatterplot with the given data, it is attached to the answer.

To be able to use the parametric linear correlation you can use the parametric test (Person) or the nonparametric test Spearman. For Person, you need your variables to have a bivariate normal distribution. Since one of the variables is a discrete variable (ratio of car crashes) and the sample is way too small to make an approximation to a normal distribution, the best test to use is Spearman's rank correlation.

This correlation coefficient (rs) takes values from -1 to 1

If rs = -1 this means that there is a negative correlation between the variables

If rs= 1 this means there is a positive correlation between the variables

If rs =0 then there is no correlation between the variables.

The hypothesis is:

H₀: There is no linear association between X₁ and X₂

H₁: There is a linear association between X₁ and X₂

α: 0.05

To calculate the Spearman's correlation coefficient you have to assign ranks to the observed values of each variable, from the smallest to the highest). Then you have to calculate the difference (d)between the ranks and the square of that difference (d²). (see attachment)

The formula for the correlation coefficient is:

[tex]rs= 1 - \frac{6* (sum of d^2)}{(n-1)n(n+1)}[/tex]

[tex]rs= 1 - \frac{6* (40)}{4*5*6}[/tex]

rs= -1

For this value of the correlation coefficient, the p-value is 0.02

Since the p-value (0.02) is less than the significance level (0.05) the decision is to reject the null hypothesis. In other words, there is a linear correlation between the imported lemons and the car crash fatality ration, this means that the modification in the lemon import will affect the car crash fatality ratio.

Note: the correlation coefficient is negative, so you could say that there is a correlation between the variables and this is negative (meaning that when the lemon import increases, the car crash fatality ratio decreases)

I hope it helps!

The petrol consumption for David's journey, considering bounds, is between 6.5439 km/l and 6.8182 km/l.

To determine the petrol consumption (c) for David's journey, we use the formula c = d/p, where d is the distance (187 km) and p is the amount of petrol used (28 litres).

Firstly, we calculate the maximum and minimum possible values for c by considering the upper and lower bounds of d and p:

Maximum value of c:

Upper bound of distance (d) = 187 + 0.5 (half of the smallest decimal place in 187 km) = 187.5 km

Lower bound of petrol used (p) = 28 - 0.5 (half of the smallest decimal place in 28 litres) = 27.5 litres

Maximum c = (187.5 km) / (27.5 litres) = 6.8182 km/l

Minimum value of c:

Lower bound of distance (d) = 187 - 0.5 = 186.5 km

Upper bound of petrol used (p) = 28 + 0.5 = 28.5 litres

Minimum c = (186.5 km) / (28.5 litres) = 6.5439 km/l

Therefore, considering bounds, the petrol consumption (c) for David's journey is between 6.5439 km/l and 6.8182 km/l.