Q: A pharmaceutical company wants to estimate the mean time it takes for a new drug to relieve headache pain. A random sample of 36 patients is given the drug, and the sample mean time is 15 minutes with a sample standard deviation of 4 minutes. Construct a 95% confidence interval for the population mean time. Which of the following is the correct interval?

$$15 \pm 1.96 \times \dfrac{4}{\sqrt{36}}$$

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

$$z_0$$ = \dfrac{0.62 - 0.5}{\sqrt{0.5 \times 0.5 / 100}}$$

Q: A researcher wants to estimate the proportion of voters who support a new policy. She wants the margin of error to be at most 0.03 with 95% confidence, and has no prior estimate for $$p. $$What is the minimum sample size she should use?

$$n = \dfrac{1.96^2$ \times 0.25}{0.03^2$}$$, rounded up

Question 1

A pharmaceutical company wants to estimate the mean time it takes for a new drug to relieve headache pain. A random sample of 36 patients is given the drug, and the sample mean time is 15 minutes with a sample standard deviation of 4 minutes. Construct a 95% confidence interval for the population mean time. Which of the following is the correct interval?

Accepted Answer

$$15 \pm 1.96 	imes \dfrac{4}{\sqrt{36}}$$

Question 2

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

Accepted Answer

$$z_0$$ = \dfrac{0.62 - 0.5}{\sqrt{0.5 	imes 0.5 / 100}}$$

Question 3

A hospital administrator wants to compare the mean recovery times for two different surgical procedures. Independent samples of 20 patients for each procedure yield sample means of 8 and 10 days, with sample standard deviations of 2 and 3 days, respectively. What is the correct formula for the 95% confidence interval for the difference in means?

Accepted Answer

$$(8 - 10) \pm t_{0.025} \sqrt{\dfrac{2^2$}{20} + \dfrac{3^2$}{20}}$$

Question 4

A quality control manager wants to know if the proportion of defective products differs between two factories. In Factory A, 30 out of 200 products are defective. In Factory B, 45 out of 250 are defective. What is the correct test statistic for comparing the two proportions?

Accepted Answer

$$z_0$$ = \dfrac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})\left(\dfrac{1}{$$n_1$$} + \dfrac{1}{$$n_2$$}\right)}}$$

Question 5

A study finds that the weights of apples in an orchard are approximately normally distributed with a mean of 150 grams and a standard deviation of 20 grams. What percentage of apples are expected to weigh between 110 and 190 grams?

Accepted Answer

Approximately 95%

Question 6

A researcher wants to estimate the proportion of voters who support a new policy. She wants the margin of error to be at most 0.03 with 95% confidence, and has no prior estimate for $$p. $$What is the minimum sample size she should use?

Accepted Answer

$$n = \dfrac{1.96^2$ 	imes 0.25}{0.03^2$}$$, rounded up

Question 7

A car manufacturer claims that their new model gets an average of 35 miles per gallon. A consumer group tests 16 cars and finds a sample mean of 33 mpg with a sample standard deviation of 3 mpg. What is the value of the test statistic for testing $$H_0$$: \mu = 35$$?

Accepted Answer

$$t_0$$ = \dfrac{33 - 35}{3 / \sqrt{16}}$$

Question 8

A biologist is interested in the number of rare birds observed in a nature reserve per day. If the mean number of sightings per day is 2, what is the probability that exactly 4 birds are seen on a given day, according to the Poisson distribution?

Accepted Answer

$$P(4) = \dfrac{2^4$ e$$^{-2}$$}{4!}$$

Question 9

A teacher wants to know if a new teaching method affects student performance. She randomly assigns 10 students to use the new method and 10 to use the traditional method. After the semester, she compares the means. Which test should she use?

Accepted Answer

Two-sample t-test for independent samples

Question 10

A city council wants to know if the distribution of car colors in their city matches the national distribution. Which statistical test should they use?

Accepted Answer

Chi-square goodness-of-fit test

Question 11

A psychologist measures the correlation between hours of sleep and test scores among 15 students and finds $$r = 0.52. $$Using Table II, is this correlation statistically significant at the 0.05 level?

Accepted Answer

Yes, because $$|r| = 0.52 \u003e 0.514$$

Question 12

A company wants to predict sales ($$y) $$based on advertising expenditure ($$x). $$The least-squares regression line is $$\hat{y} = 2x + 5. If $$the observed sales for $$x = 10 is 28$$, what is the residual?

Accepted Answer

$$28 - (2 	imes 10 + 5)$$

Question 13

A random variable $$X$$ has the following probability distribution: $$P(X=0)=0.2$$, $$P(X=1)=0.5$$, $$P(X=2)=0.3. $$What is the expected value $$E[X]$$?

Accepted Answer

$$0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3$$

Question 14

A medical researcher is interested in the effect of three different diets on cholesterol levels. She randomly assigns 30 patients to one of the three diets and measures their cholesterol after 6 months. Which statistical test should she use to compare the means?

Accepted Answer

One-way ANOVA

Question 15

A sample of 25 students has a mean test score of 80 and a standard deviation of 10. What is the z-score for a student who scored 95?

Accepted Answer

$$z = \dfrac{95 - 80}{10}$$

Question 16

A genetics lab is interested in the probability of getting exactly 3 dominant offspring in 5 independent crosses, where the probability of a dominant offspring is 0.6. What is the correct binomial probability formula?

Accepted Answer

$$P(3) = \binom{5}{3} (0.6)^3$ (0.4)^2$

Question 17

A study of 1000 randomly selected people finds that 520 prefer tea over coffee. Construct a 99% confidence interval for the proportion of people who prefer tea.

Accepted Answer

$$0.52 \pm 2.575 	imes \sqrt{\dfrac{0.52 	imes 0.48}{1000}}$$

Question 18

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 more than 1400 hours, assuming a normal distribution?

Accepted Answer

$$P(Z \u003e 2) = 1 - 0.9772 = 0.0228$$

Essential Tables and Formulas for College Statistics

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