Question 1

Given a 98% confidence interval for the proportion of U.S. adults who bought a new car in the last year as $$(0.06, 0.094)$$, what is the point estimate $$\hat{p}$$?

Accepted Answer

$$\hat{p} = 0.077$$

Question 2

In a survey of 902 randomly selected U.S. adults, 702 said they voted in the most recent presidential election. Construct a 95% confidence interval for the proportion $$p of $$adults who say they voted. Which formula should you use?

Accepted Answer

$$\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$$

Question 3

A sample of 20 OCC students' study hours is given. What requirements must be met to construct a confidence interval for the mean, and does this data satisfy them?

Accepted Answer

The data should be approximately normally distributed or $$n \geq 30$$; with $$n=20$$, check for strong skew or outliers.

Question 4

A hospital samples 30 patients' blood sugar levels and wants a 95% confidence interval for the standard deviation. What assumption must be made for the interval to be valid?

Accepted Answer

The blood sugar levels are normally distributed in the population.

Question 5

Suppose 129 out of 665 Michigan high school students in a sample dropped out. Test the claim that the dropout rate is less than 21% using the p-value method. What is the null hypothesis $$H_0$$?

Accepted Answer

$$H_0$$: p = 0.21$$

Question 6

In the context of the previous hypothesis test about Michigan high school dropout rates, what does the p-value represent?

Accepted Answer

The probability of observing a sample proportion as extreme or more extreme than the one observed, assuming $$H_0 is $$true.

Question 7

You want to estimate the percentage of people who contract a side effect from drug X with a margin of error no more than 5% at a 0.01 significance level. What formula should you use to determine the minimum sample size $$n$$?

Accepted Answer

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

Question 8

A psychology study finds the following number of lies told per day: 9, 11, 13, 22, 9, 8, 8, 8, 18, 8, 17. What assumption must be made to perform a hypothesis test on the mean?

Accepted Answer

The data are a random sample from a normally distributed population.

Question 9

A random sample of 10 supermodels had a height standard deviation $$s = 1.1 in$$, while the population standard deviation for women is $$2.6 in. $$Which test should you use to test if the supermodels' standard deviation is less than the population's?

Accepted Answer

A chi-square test for variance: $$\$$chi^2$$ = \dfrac{(n-1)s^2$}{\$$sigma_0$$^2}$$

Question 10

A sample of 100 Michigan residents found 18 say winter is their favorite season; a sample of 85 Tennessee residents found 28 say winter is their favorite. Which formula gives the 95% confidence interval for the difference in proportions $$p_1$$ - $$p_2$$?

Accepted Answer

$$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1$} + \dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2$}}$$

Question 11

Researchers test the effect of color on cognitive tasks. Red group: $$n=35, \bar{x}=15.89, s=5.9. $$Blue group: $$n=36, \bar{x}=12.31, s=5.48. $$Which test should be used to compare the means?

Accepted Answer

A two-sample t-test for independent means.

Question 12

A study group and a non-study group have the following test scores (assume normality). Which test should be used to compare the variances of the two groups?

Accepted Answer

An F-test for equality of variances: $$F = \dfrac{s_1$^2}{s_2$^2}$$

Question 13

A coach tests a new protein supplement on 7 runners, recording their race times with and without the supplement. Which test is most appropriate to determine if the supplement lowers times?

Accepted Answer

A paired t-test for mean difference.

Question 14

Suppose you construct a 95% confidence interval for a mean and it is $$(2.1, 4.3). $$How should you interpret this interval?

Accepted Answer

We are 95% confident that the true population mean lies between 2.1 and 4.3.

Question 15

In a hypothesis test, you obtain a p-value of 0.03 and your significance level is $$\alpha = 0.05. $$What is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is sufficient evidence to support the alternative.

Question 16

Which of the following is a requirement for constructing a confidence interval for a population proportion?

Accepted Answer

Both $$np \geq 5$$ and $$n(1-p) \geq 5$$

Question 17

If a 95% confidence interval for a population proportion does not contain the value 0.5, what does this imply at the 5% significance level?

Accepted Answer

There is significant evidence that the population proportion is different from 0.5.

Question 18

Which of the following best describes the margin of error in a confidence interval for a proportion?

Accepted Answer

$$z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$$

Question 19

A researcher wants to decrease the width of a confidence interval for a mean. Which of the following will achieve this?

Accepted Answer

Increase the sample size.

Statistics Exam 3 Study Guide: Step-by-Step Guidance

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