Some sources report that the weights of​ full-term newborn babies in a certain town have a mean of 7 pounds and a standard deviation of 1.2 pounds and are normally distributed.a. What is the probability that one newborn baby will have a weight within 1.2 pounds of the meanlong dashthat ​is, between 5.8 and 8.2 ​pounds, or within one standard deviation of the​ mean?b. What is the probability that the average of nine ​babies' weights will be within 1.2 pounds of the​ mean; will be between 5.8 and 8.2 ​pounds?c. Explain the difference between​ (a) and​ (b).

Answer:

We conclude that the residents of Wilmington, Delaware, have more income than the national average.

Step-by-step explanation:

We are given the following in the question:  

Population mean, μ =  $50,000

Sample mean, [tex]\bar{x}[/tex] = $60,000

Sample size, n = 10

Alpha, α = 0.05

Sample standard deviation, s = $10,000

a) First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 50000\text{ dollars}\\H_A: \mu > 50000\text{ dollars}[/tex]

We use one-tailed(right) t test to perform this hypothesis.

c) Formula:

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

Putting all the values, we have

[tex]t_{stat} = \displaystyle\frac{60000 - 50000}{\frac{10000}{\sqrt{10}} } = 3.162[/tex]

Now, [tex]t_{critical} \text{ at 0.05 level of significance, 9 degree of freedom } = 1.833[/tex]

b) Rejection Rule:

If the calculated t-statistic is greater than the the critical value, we rect the null hypothesis.

Since,                  

[tex]t_{stat} > t_{critical}[/tex]

We fail to accept the null hypothesis and reject it.We accept the alternate hypothesis.

d) There is enough evidence to conclude that the residents of Wilmington, Delaware, have more income than the national average.

Answer:

0.4

Step-by-step explanation:

To calculate the probability that a maintenance operation will take more than 2 hours and 15 minutes. We can first calculate the probability that ALL maintenance operation on 70 of the units will take less than 2 hours and 15 minutes, then subtract it from 1.

So the probability of a maintenance operation that would take less than 2 hours and 15 minutes, or 135 minutes is:

[tex]P(X \leq 135, \mu = 120, \sigma = 60) = 0.6 [/tex]

So the probability that a maintenance operation will take more than 2 hours and 15 minutes is:

[tex] 1 - 0.6 = 0.4[/tex]

Answer:

The 95% (two-sided) confidence interval for the proportion of all dies that pass the probe is (0.462, 0.566).

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence interval [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

Z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].

For this problem, we have that:

356 dies were examined by an inspection probe and 183 of these passed the probe. This neabs that [tex]n = 356, \pi = \frac{183}{356} = 0.514[/tex]

95% confidence interval

So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.514 - 1.96\sqrt{\frac{0.514*0.486}{356}} = 0.462[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.514 + 1.96\sqrt{\frac{0.514*0.486}{356}}{119}} = 0.566[/tex]

The 95% (two-sided) confidence interval for the proportion of all dies that pass the probe is (0.462, 0.566).

Answer:

Step-by-step explanation:

Given

There are 45 numbers out of which 5 numbers are required to win an enormous sum of money

No of ways  in which 5 numbers can be selected out of 45 numbers.

[tex]^nC_r=\frac{n!}{(n-r)!r!}[/tex]

here n=45, r=5

[tex]^{45}C_5=\frac{45!}{40!5!}[/tex]

[tex]^{45}C_5=1221759[/tex]

out of which there is only combination foe contest winning

[tex]P=\frac{1}{1221759}[/tex]

When 100 different tickets are bought then

Probability of winning[tex]=\frac{100}{1221759}[/tex]

The probability of winning with one combination of five numbers is [tex]\( \frac{1}{1,221,759} \)[/tex] .

The probability of winning if 100 different lottery tickets are purchased is [tex]\( 1 - \left(1 - \frac{1}{1,221,759}\right)^{100} \)[/tex], approximately [tex]\( 0.0000182 \)[/tex].

Step 1

The probability of winning the lottery with one combination of five numbers can be calculated using the formula for combinations:

[tex]\[ P(\text{win with one ticket}) = \frac{1}{\binom{45}{5}} \][/tex]   Where [tex]\( \binom{45}{5} \)[/tex]represents the number of ways to choose 5 numbers from 45 without regard to the order. Calculating [tex]\( \binom{45}{5} \)[/tex] :

[tex]\[ \binom{45}{5} = \frac{45 \times 44 \times 43 \times 42 \times 41}{5 \times 4 \times 3 \times 2 \times 1} = 1,221,759 \][/tex]

So, the probability [tex]\( P(\text{win with one ticket}) \)[/tex] is:

[tex]\[ P(\text{win with one ticket}) = \frac{1}{1,221,759} \approx 8.19 \times 10^{-7} \][/tex]

If 100 different lottery tickets are purchased, the probability of winning at least once is calculated using the complement rule:

[tex]\[ P(\text{win with 100 tickets}) = 1 - \left( 1 - \frac{1}{\binom{45}{5}} \right)^{100} \][/tex]

Step 2

Substituting [tex]\( \binom{45}{5} \)[/tex] :

[tex]\[ P(\text{win with 100 tickets}) = 1 - \left( 1 - \frac{1}{1,221,759} \right)^{100} \][/tex]

[tex]\[ P(\text{win with 100 tickets}) \approx 1 - (1 - 8.19 \times 10^{-7})^{100} \][/tex]

[tex]\[ P(\text{win with 100 tickets}) \approx 1 - (0.999999181)^{100} \][/tex]

[tex]\[ P(\text{win with 100 tickets}) \approx 1 - 0.9999818 \][/tex]

[tex]\[ P(\text{win with 100 tickets}) \approx 0.0000182 \][/tex]

The probability of winning the lottery with one ticket is approximately [tex]\( 8.19 \times 10^{-7} \)[/tex], while the probability of winning if 100 different tickets are purchased increases to approximately 0.0000182. Purchasing more tickets improves the chances of winning, but it remains a very low probability event due to the large number of possible combinations.

Answer:

1) A) II and III

2) A) Critical values: r = plusminus 0.396, significant linear correlation

3) Yi= 0.41 + 0.37Xi

Step-by-step explanation:

Hello!

The objective of the linear correlation analysis is to test if there is an association between two study variables (X₁ and X₂).

Pearson's Coefficient of correlation

For Variables with a bivariate normal distribution (X₁, X₂)~N₂(μ₁; μ₂; σ₁²; σ₂²; ρ)

To do so, the study parameter is the population coefficient of correlation (ρ) - Rho- (If you were to make a graphic of the correlation line, Rho represents the slope)

Sample coefficient of correlation: r

It takes values between -1 and 1

This coefficient gives an idea of ​​the degree of correlation between the variables.

If ρ = 0 then there is no linear correlation between X₁ and X₂ Graphically, the slope is cero

If ρ < 0 then there is a negative association between X₁ and X₂ (i.e. when one variable increases the other one decreases) In a graphic, the slope of the line is negative.

If ρ > 0 then there is a positive association between X₁ and X₂ (i.e. Both variables increase and decrease together)

The closer to 1 or -1 the coefficient is, the stronger the association between variables. Using the absolute value of the correlation coefficients you can compare them, the greater the value, the stronger is the association between variables. For example, if you were to have two coefficients r₁= -0.24 and r₂= 0.67 then the absolute values are Ir₁I= 0.24 and Ir₂I= 0.67 you can see that the coefficient of the second sample is bigger than the first sample, that means that there is a stronger correlation in the second sample than the first one.

The non-parametric coefficient of correlation has the same characteristics.

1) Statements:

I: If the linear correlation coefficient for the two variables is zero, then there is no relationship between the variables. FALSE, when r=0 then there is no linear association between the two variables, this doesn't mean that there isn't any other type of association between them.

II: If the slope of the regression line is negative, then the linear correlation coefficient is negative. TRUE

The regression and correlation analyses are closely linked because for a regression equation to be reasonable, the sample points must be linked to the equation and the correlation coefficient between both variables must be large when the degree of association is high and small when The degree of association is low in addition to being independent of the units.

The regression analysis tests whether or not there is an association between both variables and the correlation analysis indicates the degree of that association.

If the slope of the regression is negative, then the correlation coefficient is negative.

III: The value of the linear correlation coefficient always lies between -1 and 1. TRUE, it is one of the characteristics of the correlation coefficient.

0.62 suggests a stronger linear relationship than a linear correlation coefficient of -0.82. FALSE, to check wich correlation coefficient shows a stronger correlation look at their absolute values, the one that is closer to 1 is the stronger, Ir₁I= 0.62 < Ir₂I= 0.82

Correct answer:

A) II and III

2) Given the linear correlation coefficient r and the sample size n, determine the critical values of r and use your finding to state whether or not the given r represents a significant linear correlation. Use a significance level of 0.05. r = 0.523, n = 25

For this, you have to use a Table of cumulative probabilities for the linear correlation coefficient. (I've used Pearson)

For a two-tailed test (H₀: ρ=0)

[tex]r_{n-2; \alpha/2}= r_{23; 0.025}=[/tex] ± 0.396

Against r = 0.523, the decision is to reject the null hypothesis. There is a linear correlation between the two study variables.

Correct answer:

A) Critical values: r = plus-minus 0.396, significant linear correlation

3) Construct a scatterplot for the given data. Check 1st attachment for Data and Scatterplot.

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary.

Equation of regression:

Yi= a + bXi

a= [tex](\frac{sum Yi}{n})[/tex]+b[tex](\frac{sum Xi}{n})[/tex]

b= [tex]\frac{sum XiYi*\frac{(sum Xi)(sum Yi)}{n} }{/(sumXi^2)-\frac{(sumXi)^2}{n} }[/tex]

Using the given Data:

∑Xi= -11

∑Xi²= 201

∑Yi= 0

∑Yi²= 176

Mean X= -1.10

Mean Y= 0

a= 0.41

b= 0.37

Yi= 0.41 + 0.37Xi

4) Managers rate empoyees acording to job performance and attitude. The results fro several randomly selected empoyees are given below.

Performance: 59; 63; 65; 69; 58; 77; 76; 69; 70; 64

Attitude: 72; 67; 78; 82; 73; 87; 92; 83; 87; 78

No question found?

I hope it helps!

Final answer:

The linear correlation coefficient ranges from -1 to 1, indicating strength and direction of a linear relationship. Statements II and III are true concerning the correlation coefficient. A calculated coefficient of 0.523 with a sample of 25 is significant given critical values of ±0.396 at a 0.05 significance level.

Explanation:

1. The linear correlation coefficient, known as the Pearson correlation coefficient and symbolized by r, provides a measure of the strength and direction of the linear relationship between two variables. Statement II is true: If the slope of the regression line is negative, the linear correlation coefficient is also negative. Statement III is accurate as well because the value of the linear correlation coefficient lies between -1 and 1, inclusive.

For Statement I, although a correlation coefficient of zero indicates no linear relationship, there could be a non-linear relationship present. Regarding Statement IV, the strength of the relationship is determined by the absolute value of the correlation coefficient, so -0.82 indicates a stronger relationship than 0.62.

Therefore, the correct answer is A) II and III.

2. Using a Table of Critical Values for the Pearson correlation coefficient or technology like a calculator's LinRegTTest function, we compare the computed value of r to the critical values at a given significance level to determine whether the correlation is significant. For r = 0.523 and n = 25, the degrees of freedom are 23 (n - 2), and by referencing a table or using software, we find the critical values at a significance level of 0.05.

The correct answer is A) Critical values: r = ±0.396, significant linear correlation, because the reported r value of 0.523 exceeds the critical value.

Answer:

The capacity is 18 transistors, 43 resistors and 4 computer chips.

Step-by-step explanation:

This problem can be solved by a system of equations.

I am going to say that:

x is the number of transistors that are made.

y is the number of resistors that are made.

z is the number of computer chips that are made.

Building the system:

A transistor requires 1 unit of copper. A resistor requires 2 units of copper. There are 104 units of copper. So [tex]x + 2y = 104[/tex]

A transistor requires 2 units of zinc. A resistor requires 3 units of zinc. A chip requires 1 unit of zing. There are 169 units of zinc. So [tex]2x + 3y + z = 169[/tex].

A transistor requires 3 units of glass. A chip requires 5 units of glass. There are 74 units of glass. So [tex]3x + 5z = 74[/tex]

We have the following system:

[tex]x + 2y = 104[/tex]

[tex]2x + 3y + z = 169[/tex]

[tex]3x + 5z = 74[/tex]

The solution is:

[tex]x = 18, y = 43, z = 4[/tex]

The capacity is 18 transistors, 43 resistors and 4 computer chips.

Answer: a) 984   b) 1068

Step-by-step explanation:

When the prior estimate of the population proportion(p) is available .

Then the formula to find the sample size  :-

[tex]n=p(1-p)(\dfrac{z^*}{E})^2[/tex]

, where E = margin of error

and z* = Critical z-value .

a) p= 0.36

E= 0.03

Critical value for 95% confidence level = z*= 1.96

Required sample size=[tex]n= 0.36(1-0.36)(\dfrac{1.960}{0.03})^2[/tex]

[tex]n= 0.36(0.64)(65.3333333333)^2[/tex]

[tex]n=(0.2304)(4268.44444444)=983.4496\approx984[/tex]

Hence, the required sample size is 984.

b) When the prior estimate of the population proportion is unavailable .

Then we use formula to find the sample size  :-

[tex]n= 0.25(\dfrac{z^*}{E})^2[/tex]

, where E = margin of error

and z* = Critical z-value

Put E= 0.03 and z*= 1.960

Required sample size =[tex]n= 0.25(\dfrac{1.960}{0.03})^2[/tex]

[tex]n= 0.25(65.3333333333)^2[/tex]

[tex]n= 0.25(4268.44444444)=1067.11111111\approx1068[/tex]  

Hence, the required sample size is 1068.

Answer:

e)  There is a 95% chance that the ecologist's interval contains the true mean weight of trees in the population with a trunk cross-sectional area of 452 in

Step-by-step explanation:

Given that an ecologist collects data on the weights and basal trunk diameters of a randomly selected sample of felled trees. The ecologist fits a simple linear regression model, predicting tree weight from the trunk's cross-sectional area. After ensuring that all assumptions of the linear model are met, the ecologist computes a 95% confidence interval for the predicted mean weight of trees with a trunk cross-sectional area of 452 inches

We can interpret the confidence interval as

we are 95% confident that for samples of trees largely drawn with 452 inches the mean of these trees would fall within this interval.

e)  There is a 95% chance that the ecologist's interval contains the true mean weight of trees in the population with a trunk cross-sectional area of 452 in

Answer:

1 year

Step-by-step explanation:

The population as a function of time is:

[tex]P(t)  =177*4^{\frac{t}{2}}[/tex]

First, find the initial population at t=0:

[tex]P(0)  =177*4^{\frac{0}{2}}\\P(0) = 177[/tex]

Then, double it:

[tex]2P(0) = 2*177 = 354[/tex]

Finally, find the value of 't' for which the population is 354:

[tex]354  =177*4^{\frac{t}{2}}\\4^{\frac{t}{2}}=2\\log(4^{\frac{t}{2}})=log(2)\\\frac{t}{2} *log(4) = log(2)\\t=2*\frac{log(2)}{log(4)}\\t=1[/tex]

The alligator population will double afer 1 year.

To find the doubling time for the population of alligators, we set the population growth function equal to twice the original population and solve for t.

To find the doubling time for the population of alligators, we need to determine the time it takes for the population to double. In this case, the population growth function is given by P(t) = 177 * 4^(t/2), where t represents the number of years from the time of introduction. To find the doubling time, we set P(t) equal to twice the original population: 2 * P(0) = P(t).

Substituting the given population growth function, we have: 2 * 177 = 177 * 4^(t/2). To solve for t, we divide both sides of the equation by 177: 2 =

Next, we take the logarithm (base 4) of both sides of the equation to solve for t: log4(2) = t/2. Multiplying both sides by 2 gives us the doubling time: t = 2 * log4(2). Use a calculator to approximate this value.

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Answer:a) 5 sides

b) 9 sides

Step-by-step explanation:

a) Since the polygon is a regular polygon, all the interior angles are equal. If one exterior angle is 72 degrees, the interior angle would be

180 - 72 = 108 degrees(sum of the angles on a straight line is 180 degrees)

The sum of the interior angles of a polygon is expressed as

180(n - 2)

Where n represents the number of sides that the polygon has. Since the interior angles are equal, then

108n = 180(n - 2)

108n = 180n - 360

180n - 108n = 360

72n = 360

n = 360/72 = 5

b) If one exterior angle is 40 degrees, the interior angle would be

180 - 40 = 140 degrees

Since the interior angles are equal, then

140n = 180(n - 2)

140n = 180n - 360

180n - 140n = 360

40n = 360

n = 360/40 = 9

Answer:

  Distribute 2 to (x + 6) and 3 to (x – 4)

Step-by-step explanation:

A good first step for an equation such as this is to eliminate parentheses. That is accomplished by using the distributive property on the left and right. That property requires each term inside parentheses be multiplied by the factor outside parentheses. The process can be described as ...

  Distribute 2 to (x + 6) and 3 to (x – 4)

_____

Comment on further steps

The result of the first step above would be ...

  2x + 12 = 3x - 12 + 5

  19 = x . . . . . . . . . . . next step: add 7-2x to both sides and collect terms

_____

Comment on answer wording

The last answer choice uses the wording "move the 6 from the left side of the equation to the right side". This sort of description is often invoked when a constant needs to be removed from one side of the equation or the other.

  There is no property of equality that supports this action.

If you want to "move" a constant, you can invoke the addition property of equality that lets you add the same number to both sides of the equation. The number you would choose to add is the additive inverse of the one you want to "move."

In the above solution we have the sum -12+5 on the right after the first step. We want to eliminate that sum, so in the next step we show adding the additive inverse of the value of that sum. The sum of -12+5 is -7, and we add 7 to make that sum into the additive identity element, 0. (Likewise, we add -2x to turn the term 2x into the additive identity element, 0.) After these additions, we have ...

  2x -2x +12 +7 = 3x -2x -12 +5 +7

  0 + 19 = x + 0

  19 = x

The relevant property of equality is the addition property of equality, which says I can add the same amount to both sides of the equation. I can't "move" anything without violating the equal sign, but I can use the addition and multiplication properties of equality to do the same thing to both sides of the equation.

Answer:

The correct answer is Distribute 2 to (x + 6) and 3 to (x – 4).

Step-by-step explanation:

Answer: the cost of one hosta is $7

the cost of one geramium is $12

Step-by-step explanation:

Let x represent the cost of one hosta.

Let y represent the cost of one geramium.

They bought their supplies from the same store. Jill spent $101 on 11 hostas and 2 geraniums. This means that

11x + 2y = 101 - - - - - - - -1

Kim spent $80 on 8 hostas and 2 geraniums. This means that

8x + 2y = 80 - - - - - - - - -2

We will eliminate y by subtracting equation 2 from equation 1, it becomes

3x = 21

x = 21/3 = 7

Substituting x = 7 into equation 2, it becomes

8 × 7 + 2y = 80

56 + 2y = 80

2y = 80 - 56 = 24

y = 24/2 = 12

Final answer:

The equation of the slant asymptote is y = 5x + 4. The equation of the slant asymptote for the given function is found by performing polynomial long division and taking the quotient without the remainder.

Explanation:

To find the equation of the slant asymptote for the function y = (5x4 + x2 + x) / (x3 - x2 + 5), we need to divide the numerator by the denominator using polynomial long division or synthetic division. The quotient, without the remainder, will give us the equation of the slant asymptote because as x approaches infinity, the remainder becomes insignificant compared to the terms in the quotient.

Step 1: Perform the division. (This will be specific to the given function.)

Step 2: The quotient (without the remainder) is the equation of the slant asymptote.

To find the equation of the slant asymptote, we need to divide the numerator (5x^4 + x^2 + x) by the denominator (x^3 - x^2 + 5) using long division.

The quotient is 5x + 4, so the equation of the slant asymptote is y = 5x + 4. It is important to note that the slant asymptote represents the behavior of the function as x approaches positive or negative infinity.

Answer:

a) -882.6296

b) The regression equation overestimated the net income.

c) -547.0104

d) The regression equation overestimated the net income.

Step-by-step explanation:

We are given the following information in the question:

The regression equation for net income is given by:

[tex]\text{Net Income} = y = 2159 + .0312(\text{Revenue})[/tex]

a)  residual for the (x, y) pair ($45,533, $2,697)

Calculated net income =

[tex]2159 + .0312(45533) = 3579.6296[/tex]

Residual = Observed net income - Calculated net income

[tex]\text{Residual} = 2697-3579.6296 = -882.6296[/tex]

b)  The regression equation overestimated the net income.

c)  residual for the (x, y) pair ($60,417, $3,497)

Calculated net income =

[tex]2159 + .0312(60417) = 4044.0104[/tex]

Residual = Observed net income - Calculated net income

[tex]\text{Residual} = 3497-4044.0104 = -547.0104[/tex]

d)  The regression equation overestimated the net income.

The answers are : (a-1) The residual for the [tex](x,y)[/tex] pair [tex](\$45,533, $2,697)[/tex] is [tex]\[\text{Residual} = -881.6136\][/tex]. (a-2) The regression equation overestimated the net income. (b-1) The residual for the x, y pair ([tex]\$60,417, $3,497[/tex]) is [tex]\[\text{Residual} = -546.9704\][/tex]. (b-2) The regression equation overestimated the net income.

1. Calculate the predicted value [tex](\(\hat{y}\))[/tex] using the regression equation:

[tex]\[ \hat{y} = 2{,}159 + 0.0312 \times x \][/tex]

2. Compute the residual by subtracting the predicted value from the actual value:

[tex]\[ \text{Residual} = y - \hat{y} \][/tex]

Part (a-1)

For the [tex]\( x, y \) pair \((45,533, 2,697)\)[/tex]

1. Calculate the predicted net income

[tex]\[ \hat{y} = 2{,}159 + 0.0312 \times 45{,}533 \][/tex]

[tex]\[ \hat{y} = 2{,}159 + 1{,}419.6136 \][/tex]

[tex]\[ \hat{y} = 3{,}578.6136 \][/tex]

2. Compute the residual

[tex]\[ \text{Residual} = y - \hat{y} \][/tex]

[tex]\[ \text{Residual} = 2{,}697 - 3{,}578.6136 \][/tex]

[tex]\[ \text{Residual} = -881.6136 \][/tex]

Part (a-2)

The residual is negative, which means the actual net income is less than the predicted net income. Therefore, the regression equation overestimated the net income.

Part (b-1)

For the [tex]\( x, y \) pair \((60,417, 3,497)\)[/tex]

1. Calculate the predicted net income

[tex]\[ \hat{y} = 2{,}159 + 0.0312 \times 60{,}417 \][/tex]

[tex]\[ \hat{y} = 2{,}159 + 1{,}884.9704 \][/tex]

[tex]\[ \hat{y} = 4{,}043.9704 \][/tex]

2. Compute the residual

[tex]\[ \text{Residual} = y - \hat{y} \][/tex]

[tex]\[ \text{Residual} = 3{,}497 - 4{,}043.9704 \][/tex]

[tex]\[ \text{Residual} = -546.9704 \][/tex]

Part (b-2)

The residual is negative, which means the actual net income is less than the predicted net income. Therefore, the regression equation overestimated the net income.

Answer:

Step-by-step explanation:

We will use the product rule from combinatorics.

Answer: (30.49 years, 42.31 years)

Step-by-step explanation:

The formula to find the confidence interval is given by :-

[tex]\overline{x}\pm z^*\dfrac{\sigma}{\sqrt{n}}.[/tex]

, where [tex]\overline{x}[/tex] = Sample mean

z* = Critical value.

[tex]\sigma[/tex] = Population standard deviation.

n= Sample size.

As per given , we have

[tex]\overline{x}=36.4[/tex]

[tex]\sigma=14.5[/tex]

n= 40

We know that the critical value for 99% confidence interval : z* = 2.576 (By z-table)

A 99 percent confidence interval for µ, the true mean age of guests will be :

[tex]36.4\pm (2.576)\dfrac{14.5}{\sqrt{40}}\\\\ 36.4\pm (2.576)2.29265130362\\\\=36.4\pm5.90586975813\\\\\approx36.4\pm5.91\\\\=(36.4-5.91,\ 36.4+5.91)\\\\=(30.49,\ 42.31) [/tex]

∴ a 99 percent confidence interval for µ, the true mean age of guests  = (30.49 years, 42.31 years)

The 99% confidence interval for the true mean age of guests at the ski resort, with a sample mean of 36.4 years and standard deviation of 14.5 years from 40 guests, is approximately (30.5, 42.3) years.

The question is about calculating a 99 percent confidence interval for the true mean age of guests at a ski mountain resort, where the ages have a right-skewed distribution, and the population standard deviation is 14.5 years. Given a sample mean of 36.4 years from a sample of 40 guests, we can use the following formula to calculate the confidence interval:

CI = µ ± (z * (σ / √n))

Where CI is the confidence interval, µ is the sample mean, z is the z-score corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size.

For a 99% confidence interval with a sample size of 40, we need to find the z-score that corresponds to the middle 99% of the normal distribution. The z-score for a 99% confidence level is typically 2.576. Then, the margin of error (ME) can be calculated as follows:

ME = 2.576 * (14.5 / √40) ≈ 2.576 * 2.29 ≈ 5.90

Thus, the confidence interval is:

CI = 36.4 ± 5.90

So, the 99% confidence interval for the true mean would be approximately (30.5, 42.3) years.

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Answer:

[tex]\bar d=\frac{\sum_{i=1}^n d_i}{n}=-0.001[/tex]

[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i -\bar X)^2}{n-1}}=0.0095[/tex]

-The sample is too small to make judgments about skewness or symmetry.

H0:[tex]\mu_{1}=\mu_{2}[/tex]

H1:[tex]\mu_{1} \neq \mu_{2}[/tex]

[tex]t=\frac{1.173-1.174}{\sqrt{\frac{0.1506^2}{8}+\frac{0.1495^2}{8}}}=-0.013[/tex]

[tex]p_v =2*P(t_{(14)}<-0.0133)=0.990[/tex]

So the p value is a very high value and using any significance level for example [tex]\alpha=0.05, 0,1,0.15[/tex] always [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and a we don't have a significant difference between the two means.

Step-by-step explanation:

First we need to find the difference defined as:

(Operator 1 minus Operator 2)

d1=1.326-1.323=0.003      d2=1.337-1.322=0.015

d3=1.079-1.073=0.006     d4=1.229-1.233=-0.004

d5=0.936-0.934=0.002   d6=1.009-1.019=-0.01

d7=1.179-1.184=-0.005      d8=1.289-1.304=-0.015

Now we can calculate the mean of differences given by:

[tex]\bar d=\frac{\sum_{i=1}^n d_i}{n}=-0.001[/tex]

And for the sample deviation we can use the following formula:

[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i -\bar X)^2}{n-1}}=0.0095[/tex]

Describe the distribution of these differences using words. (which one is correct)

We can plot the distribution of the differences with the folowing code in R

differences<-c(0.003,0.015,0.006,-0.004,0.002,-0.01,-0.005,-0.015)

hist(differences)

And we got the image attached. And we can see that the distribution is right skewed but we don't have anough info to provide a conclusion with just 8 differnences.

-The sample is too small to make judgments about skewness or symmetry.

Use a significance test to examine the null hypothesis that the two operators have the same mean. Give the test statistic. (Round your answer to three decimal places.)

[tex]\bar X_{1}=1.173[/tex] represent the mean for the operator 1

[tex]\bar X_{2}=1.174[/tex] represent the mean for the operator 2

[tex]s_{1}=0.1506[/tex] represent the sample standard deviation for the operator 1

[tex]s_{2}=0.1495[/tex] represent the sample standard deviation for the operator 2

[tex]n_{1}=8[/tex] sample size for the operator 1

[tex]n_{2}=8[/tex] sample size for the operator 2

t would represent the statistic (variable of interest)

Concepts and formulas to use

We need to conduct a hypothesis in order to check if the means for the two groups are the same, the system of hypothesis would be:

H0:[tex]\mu_{1}=\mu_{2}[/tex]

H1:[tex]\mu_{1} \neq \mu_{2}[/tex]

If we analyze the size for the samples both are less than 30 so for this case is better apply a t test to compare means, and the statistic is given by:

[tex]t=\frac{\bar X_{1}-\bar X_{2}}{\sqrt{\frac{s^2_{1}}{n_{1}}+\frac{s^2_{2}}{n_{2}}}}[/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.

Calculate the statistic

We can replace in formula (1) like this:

[tex]t=\frac{1.173-1.174}{\sqrt{\frac{0.1506^2}{8}+\frac{0.1495^2}{8}}}=-0.013[/tex]

Statistical decision

For this case we don't have a significance level provided [tex]\alpha[/tex], but we can calculate the p value for this test. The first step is calculate the degrees of freedom, on this case:

[tex]df=n_{1}+n_{2}-2=8+8-2=14[/tex]

Since is a bilateral test the p value would be:

[tex]p_v =2*P(t_{(14)}<-0.0133)=0.990[/tex]

So the p value is a very high value and using any significance level for example [tex]\alpha=0.05, 0,1,0.15[/tex] always [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and a we don't have a significant difference between the two means.

Answer:

To test wether this is a significant decrease we have to perform a hypothesis test on the proportions.

The null hypothesis represents the past condition (the proportion of 0.23 accidents/day is equal or bigger) and the alternative hypothesis is what we claim that is happening now (the proportion have lowered).

We want to perform the test in order to know if there is enough evidence that it has changed. The result can be:

- The null hypothesis is rejected: there is enough evidence with this sample that the rate of accidents has lowered from 0.23.

- The null hypothesis failed to be rejected: there is not enough evidence to say that the rate of accidents has lowered, although the sample proportion is lower.

Step-by-step explanation:

Numerical solution:

We have to perform a hypothesis test on proportions. We want to know if there is enough evidence to claim that the number of accidents per day has lowered from 0.23.

The null and alternative hypothesis are:

[tex]H_0: \pi\geq0.23\\\\H_1:\pi<0.23[/tex]

The significance level assumed is 0.05.

The proportion of the sample is:

[tex]p=\frac{3}{30}=0.1[/tex]

The standard deviation is calculated from the population proportion

[tex]\sigma=\sqrt{\pi(1-\pi)/N} =\sqrt{0.23*(1-0.23)/30} =0.077[/tex]

The z-value now can be calculated as

[tex]z=\frac{p-\pi+0.5/N}{\sigma}=\frac{0.10-0.23+0.5/30}{0.077}  =-1.475[/tex]

The P-value for z=-1.475 is P=0.07011. The P-value is greater than the significant level, so the effect is not significant and it failed to reject the null hypothesis.

Answer:The value of the bulldozer after 3 years is $121950

Step-by-step explanation:

We would apply the straight line depreciation method. In this method, the value of the asset(bulldozer) is reduced linearly over its useful life until it reaches its salvage value. The formula is expressed as

Annual depreciation expense =

(Cost of the asset - salvage value)/useful life of the asset.

From the given information,

Useful life = 23 years

Salvage value of the bulldozer = $14950

Cost of the new bulldozer is $138000

Therefore

Annual depreciation = (138000 - 14950)/ 23 = $5350

The value of the bulldozer at any point would be V. Therefore

5350 = (138000 - V)/ t

5350t = 138000 - V

V = 138000 - 5350t

The value of the bulldozer after 3 years would be

V = 138000 - 5350×3 = $121950

Answer:

56 NOT -56

Step-by-step explanation:

That's just the answer

Answer:

56

Step-by-step explanation:

14 - (-21) + (-31) - (-25) - (-27)

= 14 +21 -31 + 25 +27

= 56

Answer:

13/9

y =-22x+32

Step-by-step explanation:

Given that the functions f and g are differentiable for all real numbers x. The table below gives values for the functions and their first derivatives at selected values of x.

x f(x) f'(x) g(x) g'(x)

1 4 -3 5 2

2 -3 -1 4 6

3 π 8 -1 4

4 -5 unknown 0 3

a) [tex]h(x) = \frac{f(x)}{g(x)} \\h'(x) = \frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}[/tex]

(using quotient rule)

Substitute 1 for x

[tex]h'(1) = \frac{g(1)f'(1)-f(1)g'(1)}{(g(1))^2}\\=\frac{9-(-4)}{9} \\=\frac{13}{9}[/tex]

b) [tex]r(x) = f(x) g(x)\\r'(x) = f(x) g'(x)+g(x)f'(x)[/tex]

when [tex]x =2, r(x) = r(2) = f(2) g(2) = -12[/tex]

point of contact is (2,-12)

Slope of tangent =[tex]r'(2) = f(2) g'(2)+g(2)f'(2)\\=-18+(-4) \\=-22[/tex]

Using point slope form, tangent is

[tex]y+12 = -22(x-2)\\y = -22x +32[/tex]

a. h'(1) = -23/25.

b. The equation of the tangent line to r(x) = f(x)g(x) at x = 2 is y = -22x + 32.

a. To find h'(1), you need to apply the quotient rule.

The quotient rule states that if you have a function h(x) = f(x) / g(x), then the derivative h'(x) is given by:

h'(x) = (f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2

In your case, h(x) = f(x) / g(x), and you want to find h'(1), so:x = 1

f(1) = 4

f'(1) = -3

g(1) = 5

g'(1) = 2

Now, plug these values into the quotient rule formula:

h'(1) = (f'(1) * g(1) - f(1) * g'(1)) / (g(1))^2

h'(1) = (-3 * 5 - 4 * 2) / (5^2)

h'(1) = (-15 - 8) / 25

h'(1) = -23 / 25

So, h'(1) = -23/25.

b. To find the equation of the tangent line to r(x) = f(x)g(x) at x = 2, you need to follow these steps:

Find r(2):

r(2) = f(2) * g(2)

r(2) = (-3) * 4

r(2) = -12

Find r'(x) using the product rule. The product rule states that if you have a function r(x) = f(x) * g(x), then the derivative r'(x) is given by:

r'(x) = f(x) * g'(x) + f'(x) * g(x)

Plugging in the given values:

r'(2) = (-3) * 6 + (-1) * 4

r'(2) = -18 - 4

r'(2) = -22

Now, you have r(2) and r'(2). You can use the point-slope form of a line to find the equation of the tangent line. The point-slope form is:

y - y1 = m(x - x1)

Where (x1, y1) is the point on the curve (in this case, (2, -12)), and m is the slope (in this case, m = r'(2)).

Plug in the values:

y - (-12) = -22(x - 2)

Simplify and solve for y:

y + 12 = -22(x - 2)

y + 12 = -22x + 44

y = -22x + 44 - 12

y = -22x + 32

So, the equation of the tangent line to r(x) at x = 2 is y = -22x + 32.

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Answer: the cost of one daylilies is $4

the cost of one ivy is $9

Step-by-step explanation:

Let x represent the cost of one daylilies.

Let y represent the cost of one ivy.

Nicole and Kim bought their supplies from the same store. Nicole spent $99 on 9 daylilies and 7 pots of ivy. This means that

9x + 7y = 99 - - - - - - - -1

Kim spent $144 on 9 daylilies and 12 pots of ivy. This means that

9x + 12y = 144 - - - - - - - - -2

We will eliminate x by subtracting equation 2 from equation 1, it becomes

- 5y = - 45

y = - 45/-5 = 9

Substituting y = 9 into equation 2, it becomes

9x + 12 × 9 = 144

9x + 108 = 144

9x = 144 - 108 = 36

x = 36/9 = 4

Answer:

[tex]P(0.26 \leq p \leq 0.43)=0.7204-0.0032=0.7172[/tex]

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 population proportion have the following distribution

[tex]p \sim N(p=0.4,\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.4(1-0.4)}{91}}=0.0514)[/tex]

And we can solve the problem using the z score on this case given by:

[tex]z=\frac{p_o -p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]

We are interested on this probability:

[tex]P(0.26 \leq p \leq 0.43)[/tex]

And we can use the z score formula, and we got this:

[tex]P(\frac{0.26 -0.4}{\sqrt{\frac{0.4(1-0.4)}{91}}} \leq Z \leq \frac{0.43 -0.4}{\sqrt{\frac{0.4(1-0.4)}{91}}})[/tex]

[tex]P(-2.726 \leq Z \leq 0.584)[/tex]

And we can find this probability like this:

[tex]P(-2.726 \leq Z \leq 0.584)=P(Z<0.584)-P(Z<-2.726)=0.7204-0.0032=0.7172[/tex]

H0: p1 − p2 ≤ 0; HA: p1 − p2 > 0 is the competing hypotheses to test the claim that there is an increase in the proportion of people who marry outside their race or ethnicity. The correct option is option A.

A hypothesis is a put forth explanation of a phenomenon (plural: hypotheses). A hypothesis must be testable according to the scientific method for it to be considered a scientific hypothesis. Scientific hypotheses are typically based on prior observations that cannot be adequately explained by the current body of knowledge.

H0:

p1 − p2 = 0;

HA: p1 −

p2 ≠ 0

H0:

p1 − p2 ≥ 0;

HA: p1 −

p2 < 0

H0:

p1 − p2 ≤ 0;

HA: p1 −

p2 > 0

H0: p1 − p2 ≤ 0; HA: p1 − p2 > 0

Therefore, the correct option is option A.

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Answer:

Cyril's adjusted gross income is $12000.

Step-by-step explanation:

First, we have that the adjusted gross income is given by the sum of the income that is not taxable.  

For our case, we have the following data:

Social security:  12000 (This is not taxable)

Wages:  5000 (This is taxable)

Interest and dividends:  4000 (This is taxable)

Unemployment compensation:  3000 (This is taxable)

municipal bond interest: $1,500 (This is not taxable)

Then, adding up the income that's deductible, we have that Cyril's adjusted gross income is:

AGI=5000+4000+3000=$12000

Answer:

Arc length [tex]=\int_0^{\pi} \sqrt{1+[(4.5sin(4.5x))]^2}\ dx[/tex]

Arc length [tex]=9.75053[/tex]

Step-by-step explanation:

The arc length of the curve is given by [tex]\int_a^b \sqrt{1+[f'(x)]^2}\ dx[/tex]

Here, [tex]f(x)=\int_0^{4.5x}sin(t) \ dt[/tex] interval [tex][0, \pi][/tex]

Now, [tex]f'(x)=\frac{\mathrm{d} }{\mathrm{d} x} \int_0^{4.5x}sin(t) \ dt[/tex]

[tex]f'(x)=\frac{\mathrm{d} }{\mathrm{d} x}\left ( [-cos(t)]_0^{4.5x} \right )[/tex]

[tex]f'(x)=\frac{\mathrm{d} }{\mathrm{d} x}\left ( -cos(4.5x)+1 \right )[/tex]

[tex]f'(x)=4.5sin(4.5x)[/tex]

Now, the arc length is [tex]\int_0^{\pi} \sqrt{1+[f'(x)]^2}\ dx[/tex]

[tex]\int_0^{\pi} \sqrt{1+[(4.5sin(4.5x))]^2}\ dx[/tex]

After solving, Arc length [tex]=9.75053[/tex]

Stokes' theorem equates the line integral of [tex]\vec F[/tex] along the curve to the surface integral of the curl of [tex]\vec F[/tex] over any surface with the given curve as its boundary. The simplest such surface is the triangle with vertices (1,0,1), (0,1,0), and (0,0,1).

Parameterize this triangle (call it [tex]T[/tex]) by

[tex]\vec s(u,v)=(1-v)((1-u)(1,0,1)+u(0,1,0))+v(0,0,1)[/tex]

[tex]\vec s(u,v)=((1-u)(1-v),u(1-v),1-u+uv)[/tex]

with [tex]0\le u\le1[/tex] and [tex]0\le v\le1[/tex]. Take the normal vector to [tex]T[/tex] to be

[tex]\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial v}=(0,1-v,1-v)[/tex]

Divide this vector by its norm to get the unit normal vector. Note that this assumes a "positive" orientation, so that the boundary of [tex]T[/tex] is traversed in the counterclockwise direction when viewed from above.

Compute the curl of [tex]\vec F[/tex]:

[tex]\vec F=(2x,2y,2x+2z)\implies\mathrm{curl}\vec F=(0,-2,0)[/tex]

Then by Stokes' theorem,

[tex]\displaystyle\int_{\partial T}\vec F\cdot\mathrm d\vec r=\iint_T\mathrm{curl}\vec F\cdot\mathrm d\vec S[/tex]

where

[tex]\mathrm d\vec S=\dfrac{\frac{\partial\vec s}{\partial u}\times\frac{\partial\vec s}{\partial v}}{\left\|\frac{\partial\vec s}{\partial u}\times\frac{\partial\vec s}{\partial v}\right\|}\,\mathrm dS[/tex]

[tex]\mathrm d\vec S=\dfrac{\frac{\partial\vec s}{\partial u}\times\frac{\partial\vec s}{\partial v}}{\left\|\frac{\partial\vec s}{\partial u}\times\frac{\partial\vec s}{\partial v}\right\|}\left\|\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial v}\right\|\,\mathrm du\,\mathrm dv[/tex]

[tex]\mathrm d\vec S=\left(\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial v}\right)\,\mathrm du\,\mathrm dv[/tex]

The integral thus reduces to

[tex]\displaystyle\int_0^1\int_0^1(0,-2,0)\cdot(0,1-v,1-v)\,\mathrm du\,\mathrm dv=\int_0^12(v-1)\,\mathrm dv=\boxed{-1}[/tex]

The question requires the use of Stokes' theorem to compute the integral of a given vector field around a defined curve. This involves calculating the curl of the vector field and a surface integral. However, specific computations were not provided.

The question is asking for the use of the Stokes' theorem to compute the integral of the vector field F=(2x,2y,2x+2z)F=(2x,2y,2x+2z) around the curve consisting of the straight lines joining the points (1,0,1), (0,1,0) and (0,0,1). In Stokes theorem, we're interested in the curl of the vector field and the surface integral of that curl over some surface S, with orientation determined by a unit normal vector. The problem also requires the computation of the curl of F which is a vector field whose components are the partial derivatives of the components of F. However, with this given information, we're unable to proceed with computations as detailed integral computations weren't provided. Therefore, a complete answer can't be provided at this time.

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Answer:

[tex]C(T(x))=1.15x-5[/tex] and [tex]T(C(x))=1.15x-5.75[/tex]

C(T(x)) represents the conditions of the coupon.

Step-by-step explanation:

The price after the coupon reduction is represented by the function

[tex]C(x)=x-5[/tex]

The price after the tip is applied is represented by the function

[tex]T(x)=1.15x[/tex]

We need to find the composite functions C(T(x)) and T(C(x)).

[tex]C(T(x))=C(1.15x)[/tex]            [tex][\because T(x)=1.15x][/tex]

[tex]C(T(x))=1.15x-5[/tex]            [tex][\because C(x)=x-5][/tex]

This function represents that 15% tip will be added first after that $5 is subtracted.

Similarly,

[tex]T(C(x))=T(x-5)[/tex]           [tex][\because C(x)=x-5][/tex]

[tex]T(C(x))=1.15(x-5)[/tex]            [tex][\because T(x)=1.15x][/tex]

[tex]T(C(x))=1.15x-5.75[/tex]

This function represents that $5 is subtracted first after that 15% tip will be added.

It is given that a coupon for $5 off any lunch price states that a 15% tip will be added to the price before the $5 is subtracted.

It means 15% tip will be added first after that $5 is subtracted. So, C(T(x)) represents the conditions of the coupon.

The functions C(T(x)) and T(C(x)) represent the application of a tip and a coupon to a lunch price, respectively, in different order. The function that correctly represents the specific conditions given by the coupon in the problem statement is T(C(x)).

The composite function C(T(x)) is found by substituting T(x) into the function C(x). So, C(T(x)) = T(x) - 5 = 1.15x - 5.

The composite function T(C(x)) is calculated by substituting C(x) into the function T(x). So, T(C(x)) = 1.15 * (x - 5) = 1.15x - 5.75.

In the context of the coupon conditions, the right composite function is T(C(x)). This composite function first applies the $5 coupon reduction (C(x)) and then the 15% tip (T(x)), which exactly follows the procedure described by the coupon.

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Answer:

n=237

Step-by-step explanation:

Previous concepts

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".

Assuming the X follows a normal distribution

[tex]X \sim N(\mu, \sigma=112.2)[/tex]

We know that the margin of error for a confidence interval is given by:

[tex]Me=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex]   (1)

The next step would be find the value of [tex]\z_{\alpha/2}[/tex], [tex]\alpha=1-0.96=0.04[/tex] and [tex]\alpha/2=0.02[/tex]

Using the normal standard table, excel or a calculator we see that:

[tex]z_{\alpha/2}=2.054[/tex]

If we solve for n from formula (1) we got:

[tex]\sqrt{n}=\frac{z_{\alpha/2} \sigma}{Me}[/tex]

[tex]n=(\frac{z_{\alpha/2} \sigma}{Me})^2[/tex]

And we have everything to replace into the formula:

[tex]n=(\frac{2.054(112.2)}{15})^2 =236.05[/tex]

And if we round up the answer we see that the value of n to ensure the margin of error required [tex]\pm=15 min[/tex] is n=237.

Answer:

B. 0.0918

Step-by-step explanation:

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.

In this problem, we have that:

[tex]\mu = 18, \sigma = 1.5[/tex]

What proportion of batteries would be expected to last less than 16 hours?

This is the pvalue of Z when X = 16. So:

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

[tex]Z = \frac{16 - 18}{1.5}[/tex]

[tex]Z = -1.33[/tex]

[tex]Z = -1.33[/tex] has a pvalue of 0.0918.

So the correct answer is:

B. 0.0918

Answer: the correct option is B

Step-by-step explanation:

the durations of the batteries are normally distributed, we would apply the formula for normal distribution which is expressed as

z = (x - u)/s

Where

x = durations of the batteries in hours

u = mean time

s = standard deviation

From the information given,

u = 18 hours

s = 1.5 hours

We want to find the proportion or probability of batteries would be expected to last less than 16 hours. It is expressed as

P(x lesser than 16)

For x = 16,

z = (16 - 18)/1.5 = - 1.33

Looking at the normal distribution table, the probability corresponding to the z score is 0.09176

Approximately 0.0918