Q: Which formula should be used to calculate the conditional probability of event A given event B?

$$P(A|B) = \frac{P(A \text{ AND } B)}{P(B)}$$

Q: Suppose scores on exams in statistics are normally distributed with unknown mean and population standard deviation of 3 points. A random sample of 36 scores has a sample mean of 67. What is the 90% confidence interval for the population mean?

$$67 \pm 1.645 \times \frac{3}{\sqrt{36}}$$

Q: What is the formula for the error bound for a population mean when the population standard deviation is known?

$$z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$$

Question 1

Which formula should be used to calculate the conditional probability of event A given event B?

Accepted Answer

$$P(A|B) = \frac{P(A 	ext{ AND } B)}{P(B)}$$

Question 2

A binomial random variable $$X$$ counts the number of successes in $$n$$ independent trials with probability $$p of $$success. What is the formula for the standard deviation of $$X$$?

Accepted Answer

$$\sigma = \sqrt{npq}$$

Question 3

A hospital surveyed 70 patients about their wait times. The sample mean was 1.5 hours and the sample standard deviation was 0.5 hours. Which distribution should be used to construct a confidence interval for the mean wait time?

Accepted Answer

Student's t-distribution

Question 4

Suppose a marketing company surveyed 200 households and found that in 120 of them, the woman made the majority of purchasing decisions. What is the point estimate for the true proportion of such households?

Accepted Answer

$$\frac{120}{200}$$

Question 5

Given a binomial distribution $$X \sim B(n, p)$$, what is the mean $$\mu$$?

Accepted Answer

$$\mu = np$$

Question 6

For a geometric distribution $$X \sim G(p)$$, what is the formula for the mean $$\mu$$?

Accepted Answer

$$\mu = \frac{1}{p}$$

Question 7

A consumer group suspects that the proportion of households with three cell phones is 30%. A company surveys 150 households and finds 43 have three cell phones. What is the correct null hypothesis for this test?

Accepted Answer

$$H_0$$: p = 0.30$$

Question 8

If the number of degrees of freedom for a chi-square distribution is 16, what is the population mean?

Accepted Answer

$$\mu = 16$$

Question 9

For a uniform distribution $$X \sim U(a, b)$$, what is the formula for the standard deviation $$\sigma$$?

Accepted Answer

$$\sigma = \sqrt{\frac{(b-a)^2$}{12}}$$

Question 10

A teacher predicts the distribution of grades as follows: A: 0.25, B: 0.30, C: 0.35, D: 0.10. In a class of 20, the observed frequencies are: A: 7, B: 7, C: 5, D: 1. What is the expected frequency for grade C?

Accepted Answer

$$0.35 	imes 20 = 7$$

Question 11

What is the formula for the chi-square test statistic in a goodness-of-fit test?

Accepted Answer

$$\sum \frac{(O - E)^2$}{E}$$

Question 12

Suppose scores on exams in statistics are normally distributed with unknown mean and population standard deviation of 3 points. A random sample of 36 scores has a sample mean of 67. What is the 90% confidence interval for the population mean?

Accepted Answer

$$67 \pm 1.645 	imes \frac{3}{\sqrt{36}}$$

Question 13

In hypothesis testing, what does the p-value represent?

Accepted Answer

The probability of obtaining a result at least as extreme as the observed result, assuming the null hypothesis is true.

Question 14

For the exponential distribution $$X \sim Exp(m)$$, what is the probability density function?

Accepted Answer

$$f(x) = m e$$^{-mx}$$

Question 15

A newly hired engineer asks 10 friends about their work weeks: 70, 50, 55, 60, 65, 60, 60, 60, 55, 60. What is the sample mean?

Accepted Answer

$$\frac{70 + 50 + 55 + 60 + 65 + 60 + 60 + 60 + 55 + 60}{10}$$

Question 16

If the observed frequencies for grades are A: 7, B: 7, C: 5, D: 1 and the expected frequencies are A: 5, B: 6, C: 7, D: 2, what is the chi-square statistic for grade A?

Accepted Answer

$$\frac{(7 - 5)^2$}{5}$$

Question 17

What is the formula for the error bound for a population mean when the population standard deviation is known?

Accepted Answer

$$z_{\alpha/2} 	imes \frac{\sigma}{\sqrt{n}}$$

Question 18

For the Central Limit Theorem, what is the standard deviation of the sampling distribution of the mean $$\bar{X}$$?

Accepted Answer

$$\frac{\sigma}{\sqrt{n}}$$

Question 19

In a two-sample hypothesis test for means with unknown population standard deviations, which calculator command should be used?

Accepted Answer

2-SampTTest

Chi-Square Goodness of Fit Test for Grade Distributions

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