Question 1

Suppose the time (in minutes) it takes to assemble a product is uniformly distributed between 20 and 40 minutes. What is the probability that a randomly selected assembly takes more than 30 minutes?

Accepted Answer

$$P(X \u003e 30) = \frac{40 - 30}{40 - 20} = \frac{10}{20} = 0.5$$

Question 2

A standardized test has scores that are normally distributed with mean $$\mu = 500$$ and standard deviation $$\sigma = 100. $$What is the probability that a randomly selected student scores above 650?

Accepted Answer

$$z = \frac{650 - 500}{100} = 1.5;\ P(X \u003e 650) = 1 - P(Z \u003c 1.5) = 1 - 0.9332 = 0.0668$$

Question 3

If the 90th percentile of a standard normal distribution is needed, what is the corresponding $$z-$$score?

Accepted Answer

$$z = 1.28$$

Question 4

A population has mean $$\mu = 80$$ and standard deviation $$\sigma = 12. If a $$random sample of size $$n = 36 is $$taken, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{12}{6} = 2$$

Question 5

Why do we construct confidence intervals in statistics?

Accepted Answer

To estimate an unknown population parameter with an associated level of confidence

Question 6

A 95% confidence interval for a population mean is calculated as (52, 60). Which of the following is the correct interpretation?

Accepted Answer

We are 95% confident that the true population mean lies between 52 and 60.

Question 7

A researcher wants to estimate the mean systolic blood pressure of adults with a margin of error of 4 mmHg at 95% confidence. If the population standard deviation is estimated to be 20 mmHg, what is the minimum sample size required?

Accepted Answer

$$n = \left( \frac{z_{0.025} \cdot \sigma}{E} \$$right)^2$$ = \left( \frac{1.96 	imes 20}{4} \$$right)^2$$ = (9.8)^2$ = 96.04 \approx 97$$

Question 8

Which of the following best describes the null hypothesis ($$H_0) in $$hypothesis testing?

Accepted Answer

It is a statement of no effect or no difference, assumed true until evidence suggests otherwise.

Question 9

A company claims that the average lifetime of its light bulbs is at least 1200 hours. A consumer group tests this claim using a significance level of $$\alpha = 0.05. $$What does the significance level represent in this context?

Accepted Answer

The probability of rejecting the null hypothesis when it is actually true.

Question 10

In a hypothesis test, which of the following best describes a Type II error?

Accepted Answer

Failing to reject the null hypothesis when the alternative hypothesis is true.

Question 11

A clinical trial compares the mean recovery times for two independent groups: one receiving a new drug and one receiving a placebo. Population variances are unknown and not assumed equal. Which test should be used?

Accepted Answer

Welch's t-test for independent samples

Question 12

Which formula is used to compute the pooled standard deviation ($$s_p) $$for two independent samples with equal population variances?

Accepted Answer

$$s_p = \sqrt{\frac{(n_1$ - 1)s_1$^2 + (n_2$ - 1)s_2$^2}{n_1$ + n_2$ - 2}}$$

Question 13

A researcher conducts a paired t-test to compare pre- and post-treatment cholesterol levels in the same patients. Which of the following is the correct null hypothesis?

Accepted Answer

$$H_0$$: \mu_d = 0$$

Question 14

A sample of 25 students has a mean test score of 78 and a sample standard deviation of 10. Construct a 95% confidence interval for the population mean. What is the correct interval?

Accepted Answer

$$78 \pm t_{0.025,24} 	imes \frac{10}{\sqrt{25}} = 78 \pm 2.064 	imes 2 = 78 \pm 4.128 = (73.87, 82.13)$$

Question 15

Which of the following statements about the power of a statistical test is correct?

Accepted Answer

It is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true.

Question 16

A hospital administrator wants to compare the mean length of stay for patients in two different wards. The samples are independent and population variances are assumed equal. Which test should be used?

Accepted Answer

Pooled t-test for independent samples

Question 17

A sample mean is 105, the population mean is 100, the sample standard deviation is 15, and the sample size is 25. What is the value of the test statistic for testing $$H_0$$: \mu = 100$$?

Accepted Answer

$$t = \frac{105 - 100}{15/\sqrt{25}} = \frac{5}{3} = 1.67$$

Question 18

Which of the following best describes the classical approach to hypothesis testing?

Accepted Answer

Compare the test statistic to the critical value; reject $$H_0 if $$the test statistic falls in the critical region.

Exam 2 Study Guide: Probability Distributions, Sampling, Confidence Intervals, and Hypothesis Testing

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