Q: A random sample of 12 sliders has a mean speed of 54.2 mph and a sample standard deviation of 3.1 mph. What is the margin of error for a 95% confidence interval for the mean speed? (Use $$ t^* = 2.201 $$ for 11 degrees of freedom.)

$$ 2.201 \times \frac{3.1}{\sqrt{12}} = 1.97 $$ mph

Question 1

Which of the following best describes a point estimate for a population proportion?

Accepted Answer

A statistic that estimates the value of a population parameter using sample data.

Question 2

Suppose a survey of 250 adults finds that 120 support a new policy. What is the point estimate for the population proportion supporting the policy?

Accepted Answer

$$ \frac{120}{250} = 0.48 $$

Question 3

Which formula is used to calculate the standard error of the sample proportion $$ \hat{p} $$?

Accepted Answer

$$ \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$

Question 4

A 95% confidence interval for a population proportion is constructed as $$ \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} . $$What does $$ z^* $$ represent?

Accepted Answer

The critical value from the standard normal distribution for the desired confidence level.

Question 5

If you increase the level of confidence from 90% to 99%, what happens to the width of the confidence interval for a population proportion?

Accepted Answer

The width increases because the critical value $$ z^* $$ increases.

Question 6

Which of the following is the correct formula for determining the sample size $$ n $$ needed to estimate a population proportion within a specified margin of error $$ E  at $$confidence level $$ (1-\alpha) $$?

Accepted Answer

$$ n = \left( \frac{z^*}{E} \$$right)^2$$ \hat{p}(1-\hat{p}) $$

Question 7

When constructing a confidence interval for a population mean and the population standard deviation is unknown, which distribution should be used?

Accepted Answer

Student's $$ t -$$distribution

Question 8

Which property is true about the $$ t -$$distribution as the degrees of freedom increase?

Accepted Answer

It approaches the standard normal distribution.

Question 9

A random sample of 12 sliders has a mean speed of 54.2 mph and a sample standard deviation of 3.1 mph. What is the margin of error for a 95% confidence interval for the mean speed? (Use $$ t^* = 2.201 $$ for 11 degrees of freedom.)

Accepted Answer

$$ 2.201 	imes \frac{3.1}{\sqrt{12}} = 1.97 $$ mph

Question 10

Which formula is used to construct a $$(1-\alpha)100\%$$ confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

$$ \bar{x} \pm t^* \frac{s}{\sqrt{n}} $$

Question 11

Which of the following is NOT a property of the $$ t -$$distribution?

Accepted Answer

It is always narrower than the standard normal distribution.

Question 12

A confidence interval for a population variance is constructed using which distribution?

Accepted Answer

Chi-square distribution

Question 13

Which formula gives the lower bound for a $$(1-\alpha)100\%$$ confidence interval for a population variance $$ \$$sigma^2$$ $$?

Accepted Answer

$$ \frac{(n-1)s^2$}{\$$chi^2$$_{\alpha/2}} $$

Question 14

Which statement about bootstrapping is correct?

Accepted Answer

Bootstrapping uses resampling with replacement from the original sample to estimate parameters.

Question 15

In the bootstrap method, what is the purpose of generating many resamples from the original sample?

Accepted Answer

To approximate the sampling distribution of the statistic.

Question 16

Which of the following is a correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

If we take many samples and construct intervals, about 95% of them will contain the population mean.

Question 17

A researcher wants to estimate the mean amount spent on Valentine's Day gifts with a margin of error of $$ E = 2 $$ and a standard deviation estimate of $$ s = 8 . $$What sample size is needed for a 95% confidence interval? (Use $$ z^* = 1.96 )$$

Accepted Answer

$$ n = \left( \frac{1.96 	imes 8}{2} \$$right)^2$$ = 61.47  ($$rounded up to 62)

Question 18

Which of the following is a necessary condition for using the $$ t -$$distribution to construct a confidence interval for a population mean?

Accepted Answer

The sample must be random and the population must be approximately normal if the sample size is small.

Question 19

Which of the following statements about the margin of error is correct?

Accepted Answer

The margin of error decreases as the sample size increases.

Estimating the Value of a Parameter: Confidence Intervals and Bootstrapping

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