Question 1

A random sample of 15 U.S. households is taken. If the probability that a household has a DVD player is 0.7, what is the probability that exactly 10 households have a DVD player? Use the binomial probability formula.

Accepted Answer

$$P(X = 10) = \binom{15}{10} (0.7$$)^{10}$$ (0.3$$)^{5}$$

Question 2

In the hypothesis test $$H_0$$: \mu = 12$$, H_1$: \mu \u003c 12$$, a p-value of 0.003 is found at a 1% significance level. What is the correct interpretation?

Accepted Answer

The population mean is below 12.

Question 3

Which theorem states that as sample size $$n$$ increases, the distribution of sample means approaches a normal distribution, regardless of the population’s original distribution?

Accepted Answer

Central Limit Theorem

Question 4

When calculating the sample size $$n$$ needed to achieve a certain margin of error for a proportion, which value of $$\hat{p}$$ will lead to the largest required sample size?

Accepted Answer

$$\hat{p} = 0.45$$

Question 5

To increase the precision of a confidence interval for a population mean, which action should you take?

Accepted Answer

Increase the sample size

Question 6

A random sample of 4000 Americans found that 2200 bought a present for Valentine’s Day. What is the point estimate for the proportion of Americans who celebrate Valentine’s Day?

Accepted Answer

$$\hat{p} = \frac{2200}{4000} = 0.55$$

Question 7

SAT Reading scores are normally distributed with mean $$\mu = 510$$ and standard deviation $$\sigma = 70. $$What is the probability that a randomly selected student scored below 490?

Accepted Answer

$$P(X \u003c 490) = P\left(Z \u003c \frac{490 - 510}{70}\right) = P(Z \u003c -0.286)$$

Question 8

A 99% confidence interval for a mean is constructed with a total length of 9. What is the margin of error?

Accepted Answer

4.5

Question 9

Given the following frequencies for goals scored in World Cup finals: 1 goal (3 times), 2 goals (3 times), 3 goals (1 time), 5 goals (2 times), 6 goals (1 time). What is the probability that in a randomly selected final, teams score 6 goals?

Accepted Answer

$$P(X = 6) = \frac{1}{10}$$

Question 10

A retailer claims the mean waiting time is less than 5 minutes. A sample of 20 waiting times has mean $$\bar{x} = 3.5$$ and $$s = 2.1. At$$ $$\alpha = 0.01$$, what is the test statistic for the hypothesis $$H_0$$: \mu = 5$$, H_1$: \mu \u003c 5$$?

Accepted Answer

$$t = \frac{3.5 - 5}{2.1 / \sqrt{20}} \approx -3.21$$

Question 11

A researcher wants to use the least possible sample size for a proportion estimate. If previous studies found the proportion between 0.1 and 0.15, which value should be used to calculate the sample size?

Accepted Answer

0.15

Question 12

Which of the following is NOT a condition for data to be considered normally distributed when analyzing a box-plot and a normality plot?

Accepted Answer

The data has significant skewness and outliers.

Question 13

An airline claims the no-show rate is less than 5%. In a sample of 420 reservations, 19 were no-shows. What is the sample proportion $$\hat{p}$$ rounded to the nearest thousandth?

Accepted Answer

$$\hat{p} = \frac{19}{420} \approx 0.045$$

Question 14

If you want to estimate the proportion of Americans who celebrate Valentine’s Day with a 98% confidence interval, which formula should you use for the margin of error?

Accepted Answer

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

Question 15

Which of the following best describes the Law of Large Numbers?

Accepted Answer

As the sample size increases, the sample mean approaches the population mean.

Question 16

For a binomial distribution with $$n = 15$$ and $$p = 0.7$$, what is the mean and standard deviation?

Accepted Answer

Mean: $$\mu = np = 10.5$$, Standard deviation: $$\sigma = \sqrt{np(1-p)} = \sqrt{15 	imes 0.7 	imes 0.3} \approx 1.77$$

Question 17

A hypothesis test for a population proportion $$p = 0.250 is $$conducted with $$n = 48$$ and $$\hat{p} = 0.231 at$$ $$\alpha = 0.01. $$What is the formula for the test statistic?

Accepted Answer

$$z = \frac{\hat{p} - p_0$}{\sqrt{\frac{p_0$(1-p_0$)}{n}}}$$

Question 18

If the probability that at least one household in a sample of 15 has a DVD player is required, which formula should be used?

Accepted Answer

$$P(X \geq 1) = 1 - P(X = 0)$$

Probability and Statistics Final Review Guidance

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