Q: A researcher wants to test if the mean cholesterol level in a population is different from 200 mg/dL. She collects a sample and performs a two-sided hypothesis test. Which is the correct set of null and alternative hypotheses?

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

Q: 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 $$E[X]$$?

$$0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1$$

Question 1

A pharmaceutical company wants to test whether a new drug lowers blood pressure more effectively than the current standard. They randomly assign 100 patients to receive either the new drug or the standard drug. Which statistical concept is best illustrated by the random assignment in this experiment?

Accepted Answer

Random assignment helps to control for confounding variables, allowing for causal inference.

Question 2

A researcher collects data on the heights (in cm) of 50 adult males and finds the mean is 175 cm with a standard deviation of 8 cm. Which formula should be used to calculate the z-score for a man who is 190 cm tall?

Accepted Answer

$$z = \frac{190 - 175}{8}$$

Question 3

Which of the following best describes the difference between a population parameter and a sample statistic?

Accepted Answer

A population parameter is a fixed value, while a sample statistic is a variable calculated from data.

Question 4

A hospital administrator wants to estimate the average length of stay for patients. She takes a random sample of 40 patients and calculates a 95% confidence interval for the mean. Which formula should she use to calculate the confidence interval if the population standard deviation is unknown?

Accepted Answer

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

Question 5

A study finds a correlation coefficient $$r = -0.85$$ between hours spent watching TV and GPA among college students. Which of the following is the best interpretation?

Accepted Answer

There is a strong negative linear relationship between hours spent watching TV and GPA.

Question 6

A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?

Accepted Answer

$$\frac{1}{3}$$

Question 7

A factory produces light bulbs, and the probability that a bulb is defective is 0.02. If 10 bulbs are selected at random, what is the expected number of defective bulbs?

Accepted Answer

$$10 	imes 0.02 = 0.2$$

Question 8

A sample of 100 students has a mean test score of 80 and a standard deviation of 10. What is the standard error of the mean?

Accepted Answer

$$\frac{10}{\sqrt{100}} = 1$$

Question 9

A researcher wants to test if the mean cholesterol level in a population is different from 200 mg/dL. She collects a sample and performs a two-sided hypothesis test. Which is the correct set of null and alternative hypotheses?

Accepted Answer

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

Question 10

A survey is conducted to compare the mean income of two cities. The sample means are $$\bar{x}_1 = 55,000$$ and $$\bar{x}_2 = 52,000$$, with standard deviations $$s_1 = 5$$,000$$ and s_2$ = 4,000$$, and sample sizes $$n_1 = 40$, n_2$ = 35. $$Which formula should be used to calculate the standard error for the difference in means?

Accepted Answer

$$\sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}$$

Question 11

A biologist is interested in the relationship between fertilizer amount and crop yield. She finds the regression equation $$\hat{y} = 2.5 + 0.8x$$, where $$x is $$the amount of fertilizer (kg) and $$y is $$the yield (tons). What is the predicted yield when 5 kg of fertilizer is used?

Accepted Answer

$$2.5 + 0.8 	imes 5 = 6.5$$

Question 12

A medical researcher wants to estimate the proportion of patients who respond to a new treatment. In a sample of 200 patients, 60 respond. What is the point estimate for the proportion?

Accepted Answer

$$\frac{60}{200} = 0.3$$

Question 13

A teacher wants to know if there is an association between students' preferred study method (group or solo) and their pass/fail status on an exam. Which statistical test is most appropriate?

Accepted Answer

Chi-square test of independence

Question 14

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

$$0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 1.1$$

Question 15

A sample of 25 batteries has a mean lifetime of 120 hours and a sample standard deviation of 15 hours. Assuming normality, what is the 95% confidence interval for the mean battery lifetime?

Accepted Answer

$$120 \pm t^* \frac{15}{\sqrt{25}}$$

Question 16

A university claims that at least 80% of its graduates find employment within six months. A random sample of 150 graduates shows that 110 found employment. Which hypothesis test should be used to evaluate the university's claim?

Accepted Answer

One-sample proportion z-test

Question 17

A researcher is interested in the distribution of IQ scores, which are normally distributed with mean 100 and standard deviation 15. What proportion of the population has an IQ between 85 and 115?

Accepted Answer

Approximately 68%

Question 18

A scientist wants to compare the mean weight of two species of fish. She collects independent samples and finds the following: Species A: $$\bar{x}_A = 2.1 kg$$, $$s_A = 0.3 kg$$, $$n_A = 20$$; Species B: $$\bar{x}_B = 1.8 kg$$, $$s_B = 0.4 kg$$, $$n_B = 25. $$What is the test statistic for comparing the means?

Accepted Answer

$$\frac{2.1 - 1.8}{\sqrt{\frac{0.3^2$}{20} + \frac{0.4^2$}{25}}}$$

Question 19

A student wants to summarize the distribution of exam scores for a class. Which graphical method is most appropriate for displaying the variability in the scores?

Accepted Answer

Histogram

STT 215: Core Concepts and Learning Outcomes in Introductory Statistics

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