Q: A researcher is testing whether a coin is fair. She flips it 100 times and gets 62 heads. What is the value of the test statistic for testing $$H_0$$: p = 0.5$$ using the normal approximation?

$$z = \frac{0.62 - 0.5}{\sqrt{0.5 \times 0.5 / 100}} = 2.4$$

Q: A genetics lab finds that the probability of a certain gene mutation is 0.02. In a sample of 200 individuals, what is the expected number of people with the mutation and the standard deviation?

Expected: $$4$$, SD: $$\sqrt{200 \times 0.02 \times 0.98} \approx 1.98$$

Q: A city council wants to estimate the proportion of residents in favor of a new park. What sample size is needed to estimate the proportion within $$\pm 0.04$$ with 95% confidence, assuming no prior estimate?

$$n = \frac{(1.96)^2$ \times 0.25}{0.04^2$} = 600.25$$

Q: A clinical trial compares the effectiveness of two drugs. The difference in sample means is 2.5, with a standard error of 1.2. What is the 95% confidence interval for the difference in means?

$$2.5 \pm 1.96 \times 1.2 = (0.14, 4.86)$$

Question 1

A pharmaceutical company wants to estimate the mean recovery time for a new drug. A sample of 36 patients has a mean recovery time of 8 days with a standard deviation of 3 days. Construct a 95% confidence interval for the mean recovery time. Which formula should be used, and what is the correct interval?

Accepted Answer

Use $$E = z_{\alpha/2} \frac{s}{\sqrt{n}}$$; the interval is $$8 \pm 1.96 	imes \frac{3}{6}$$, or $$(7.02, 8.98)$$

Question 2

A researcher is testing whether a coin is fair. She flips it 100 times and gets 62 heads. What is the value of the test statistic for testing $$H_0$$: p = 0.5$$ using the normal approximation?

Accepted Answer

$$z = \frac{0.62 - 0.5}{\sqrt{0.5 	imes 0.5 / 100}} = 2.4$$

Question 3

A hospital administrator wants to compare the mean waiting times for two emergency rooms. The sample means are 30 and 25 minutes, with sample standard deviations of 8 and 10 minutes, and sample sizes of 40 and 35, respectively. What is the correct formula for the standard error of the difference in means?

Accepted Answer

$$SE = \sqrt{\frac{8^2$}{40} + \frac{10^2$}{35}}$$

Question 4

A genetics lab finds that the probability of a certain gene mutation is 0.02. In a sample of 200 individuals, what is the expected number of people with the mutation and the standard deviation?

Accepted Answer

Expected: $$4$$, SD: $$\sqrt{200 	imes 0.02 	imes 0.98} \approx 1.98$$

Question 5

A quality control manager wants to know if two machines produce items with the same mean weight. She collects two independent samples. Which test statistic should she use if the population variances are unknown but assumed equal?

Accepted Answer

$$t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1$ - \mu_2$)}{\sqrt{s_p^2$ / n_1$ + s_p^2$ / n_2$}}$$

Question 6

A study finds a correlation coefficient $$r = 0.68$$ between hours studied and exam scores for $$n = 12$$ students. At $$\alpha = 0.05$$, is this correlation significant?

Accepted Answer

Yes, because $$|r| = 0.68 \u003e 0.576 ($$critical value for $$n=12 at$$ $$\alpha=0.05)$$

Question 7

A city council wants to estimate the proportion of residents in favor of a new park. What sample size is needed to estimate the proportion within $$\pm 0.04$$ with 95% confidence, assuming no prior estimate?

Accepted Answer

$$n = \frac{(1.96)^2$ 	imes 0.25}{0.04^2$} = 600.25$$

Question 8

A researcher wants to test if a new teaching method improves test scores. The null hypothesis is $$H_0$$: \mu = 75$$. The sample mean is 78, s = 10$, n = 25$. What is the value of the test statistic?

Accepted Answer

$$t = \frac{78 - 75}{10 / \sqrt{25}} = 1.5$$

Question 9

A factory produces light bulbs with a mean life of 1200 hours and a standard deviation of 100 hours. What is the probability that a randomly selected bulb lasts less than 1100 hours?

Accepted Answer

Find $$z = \frac{1100 - 1200}{100} = -1. $$From the standard normal table, $$P(X \u003c 1100) = 0.1587$$

Question 10

A clinical trial compares the effectiveness of two drugs. The difference in sample means is 2.5, with a standard error of 1.2. What is the 95% confidence interval for the difference in means?

Accepted Answer

$$2.5 \pm 1.96 	imes 1.2 = (0.14, 4.86)$$

Question 11

A study of 500 people finds that 120 have a certain medical condition. What is the 99% confidence interval for the proportion of people with the condition?

Accepted Answer

$$0.24 \pm 2.575 	imes \sqrt{0.24 	imes 0.76 / 500}$$

Question 12

A researcher wants to test if the variance of a process is greater than 4. A sample of 15 observations has a sample variance of 6. What is the value of the test statistic?

Accepted Answer

$$\$$chi^2$$ = \frac{(15 - 1) 	imes 6}{4} = 21$$

Question 13

A company wants to know if the proportion of defective products differs between two factories. Factory A: 30 defects in 400 items; Factory B: 50 defects in 600 items. What is the pooled proportion?

Accepted Answer

$$p = \frac{30 + 50}{400 + 600} = 0.08$$

Question 14

A biologist is interested in the number of rare birds seen per day in a wildlife reserve. The mean number is 0.8 per day. What is the probability that exactly 2 birds are seen on a given day?

Accepted Answer

$$P(2) = \frac{0.8^2$ e$$^{-0.8}$$}{2!}$$

Question 15

A teacher wants to know if a new curriculum affects student performance. The original claim is that the mean score is at least 70. The test results in failing to reject $$H_0. $$What is the correct nontechnical conclusion?

Accepted Answer

There is not sufficient evidence to warrant rejection of the claim that the mean score is at least 70.

Question 16

A sociologist wants to test if the distribution of favorite colors among children matches the expected distribution. Which test should be used?

Accepted Answer

Chi-square goodness-of-fit test: $$\$$chi^2$$ = \sum \frac{(O - E)^2$}{E}$$

Question 17

A study finds that the mean cholesterol level in a population is 200 mg/dL with a standard deviation of 25 mg/dL. What is the z-score for a person with a cholesterol level of 240 mg/dL?

Accepted Answer

$$z = \frac{240 - 200}{25} = 1.6$$

Essential Formulas, Tables, and Procedures for College Statistics

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드