Q: A researcher wants to test if the mean cholesterol level in a population is different from 200 mg/dL. She collects a random sample of 40 individuals. Which is the correct null and alternative hypothesis for a two-tailed t test?

$$H_0$$: \mu = 200$$; H_1$: \mu \neq 200$$

Question 1

A pharmaceutical company wants to estimate the proportion of patients who experience side effects from a new drug. In a random sample of 200 patients, 30 reported side effects. Construct a 95% confidence interval for the true proportion of patients experiencing side effects. Which formula should be used for the margin of error?

Accepted Answer

$$z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 2

Which of the following is NOT an assumption required for constructing a confidence interval for a single population mean using the t-distribution?

Accepted Answer

The population standard deviation is known.

Question 3

A researcher is testing whether a new teaching method improves test scores. She collects paired data (before and after scores) from 25 students. Which hypothesis test should she use?

Accepted Answer

Two sample t test for dependent means (matched pairs)

Question 4

Suppose you want to determine the sample size needed to estimate a population proportion with a margin of error $$E = 0.03 at a 95$$% confidence level, and you have no prior estimate for $$\hat{p}. $$Which formula should you use?

Accepted Answer

$$n = 0.25 \left( \frac{z_{\alpha/2}}{E} \$$right)^2$$

Question 5

A hospital administrator wants to compare the mean recovery times of two different physical therapy treatments. The samples are independent and have unequal variances. Which test statistic should be used?

Accepted Answer

$$t_0$$ = \frac{(\bar{x}_1 - \bar{x}_2) - (\$$mu_1$$ - \$$mu_2$$)}{\sqrt{\frac{$$s_1^2$$}{$$n_1$$} + \frac{$$s_2^2$$}{$$n_2$$}}}$$

Question 6

Which of the following best describes a Type II error in hypothesis testing?

Accepted Answer

Failing to reject the null hypothesis when the alternative hypothesis is actually true.

Question 7

A clinical trial compares the proportion of patients who recover using two different drugs. The sample sizes are $$n_1 = 80$ and n_2$ = 90$$, with observed proportions $$\hat{p}_1 = 0.65$$ and $$\hat{p}_2 = 0.55. $$What is the pooled proportion $$\hat{p}$$ used in the two-sample z test for proportions?

Accepted Answer

$$\hat{p} = \frac{x_1$ + x_2$}{n_1$ + n_2$}$$

Question 8

Which of the following scenarios would require the use of a bootstrap confidence interval rather than a traditional t or z interval?

Accepted Answer

The population distribution is highly skewed and the sample size is small.

Question 9

A researcher wants to test if the mean cholesterol level in a population is different from 200 mg/dL. She collects a random sample of 40 individuals. Which is the correct null and alternative hypothesis for a two-tailed t test?

Accepted Answer

$$H_0$$: \mu = 200$$; H_1$: \mu 
eq 200$$

Question 10

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

Accepted Answer

If we were to take many random samples and construct a confidence interval from each, about 95% of those intervals would contain the true population mean.

Question 11

A study compares the mean blood pressure before and after a new medication in the same group of patients. The mean difference is $$\bar{d} = -5$$ mmHg, $$s_d = 8$$ mmHg, and $$n = 25. $$What is the test statistic for testing $$H_0$$: \mu_d = 0$$?

Accepted Answer

$$t_0$$ = \frac{\bar{d}}{\frac{s_d}{\sqrt{n}}}$$

Question 12

Which of the following is a correct assumption for using the two-sample t test for independent means (assuming unequal variances)?

Accepted Answer

The two samples are independent and each is obtained by simple random sampling.

Question 13

A random sample of 100 voters is taken to estimate the proportion who support a new policy. The sample proportion is $$\hat{p} = 0.6. $$What is the lower bound of the 90% confidence interval for the true proportion?

Accepted Answer

$$\hat{p} - z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 14

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

Accepted Answer

Reject the null hypothesis; the result is statistically significant.

Question 15

A scientist wants to estimate the mean weight of a species of fish in a lake. She wants the margin of error to be no more than 2 grams at a 95% confidence level. The sample standard deviation from a pilot study is 5 grams. Which formula should she use to determine the required sample size?

Accepted Answer

$$n = \left( \frac{z_{\alpha/2} \cdot s}{E} \$$right)^2$$

Question 16

Which of the following best describes the difference between statistical significance and practical significance?

Accepted Answer

Statistical significance means the result is unlikely due to chance, while practical significance considers whether the effect size is large enough to be meaningful in real life.

Question 17

A randomization test is most appropriate when:

Accepted Answer

The assumptions for traditional parametric tests are not met, such as non-normality or small sample size.

Question 18

Which of the following is the correct formula for the confidence interval for the difference between two independent means (assuming unequal variances)?

Accepted Answer

$$(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \cdot \sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}$$

Statistics Unit 3: Confidence Intervals and Hypothesis Testing (Chapters 9-11) – Study Guide

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