Q: Suppose you want to estimate the mean daily sales of a coffee shop with a margin of error $$E = 5$$ and a standard deviation $$\sigma = 20. $$What is the minimum sample size required for a 95% confidence interval? (Use $$z_{0.025} = 1.96)$$

$$n = \left( \frac{1.96 \times 20}{5} \$$right)^2$$ = 61.47$$, so $$n = 62$$

Q: A biologist collects a random sample of 7 beetles and measures their lengths. The sample mean is $$2.1 cm $$and the sample standard deviation is $$0.2 cm. $$What is the 95% confidence interval for the mean length? (Use $$t_{0.025,6} = 2.447)$$

$$2.1 \pm 2.447 \times \frac{0.2}{\sqrt{7}}$$

Q: A civil engineer wants to estimate the average compressive strength of a type of concrete. She collects a sample of $$n = 25$$ and finds a sample mean of $$3700$$ psi with a sample standard deviation of $$150$$ psi. What is the 90% confidence interval for the mean? (Use $$t_{0.05,24} = 1.711)$$

$$3700 \pm 1.711 \times \frac{150}{\sqrt{25}}$$

Question 1

A researcher is testing whether a new medication lowers blood pressure more than the current standard. The null hypothesis is $$H_0$$: \mu = 120$$ mmHg and the alternative is H_a$$: \mu \u003c 120$$ mmHg. If the researcher fails to reject H_0$ when the medication actually does lower blood pressure, what type of error has occurred?

Accepted Answer

Type II error

Question 2

Which of the following best describes the p-value in hypothesis testing?

Accepted Answer

The probability of obtaining a result as extreme or more extreme than the observed, assuming $$H_0 is $$true

Question 3

A factory alarm system is designed to detect machine failures. The null hypothesis is that the machinery is operating normally. If the alarm sounds when the machinery is actually working properly, what type of error is this?

Accepted Answer

Type I error

Question 4

Suppose you want to estimate the mean daily sales of a coffee shop with a margin of error $$E = 5$$ and a standard deviation $$\sigma = 20. $$What is the minimum sample size required for a 95% confidence interval? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$n = \left( \frac{1.96 	imes 20}{5} \$$right)^2$$ = 61.47$$, so $$n = 62$$

Question 5

Which of the following statements about the Student's t-distribution is TRUE?

Accepted Answer

It approaches the standard normal distribution as the sample size increases

Question 6

A biologist collects a random sample of 7 beetles and measures their lengths. The sample mean is $$2.1 cm $$and the sample standard deviation is $$0.2 cm. $$What is the 95% confidence interval for the mean length? (Use $$t_{0.025,6} = 2.447)$$

Accepted Answer

$$2.1 \pm 2.447 	imes \frac{0.2}{\sqrt{7}}$$

Question 7

If you decrease the significance level $$\alpha$$ from $$0.05 to$$ $$0.01 in a $$hypothesis test, what is the effect on the probability of a Type I error and the rejection region?

Accepted Answer

Type I error probability decreases; rejection region becomes smaller

Question 8

A researcher wants to estimate the average recovery time for a surgical procedure. The sample mean is $$4.2$$ days, and the 95% confidence interval is $$(3.8, 4.6). $$What is the sampling error?

Accepted Answer

$$|4.2 - 3.8| = 0.4$$

Question 9

Which of the following is NOT an assumption for constructing a confidence interval for the mean using the t-distribution?

Accepted Answer

The population standard deviation $$\sigma is $$known

Question 10

A civil engineer wants to estimate the average compressive strength of a type of concrete. She collects a sample of $$n = 25$$ and finds a sample mean of $$3700$$ psi with a sample standard deviation of $$150$$ psi. What is the 90% confidence interval for the mean? (Use $$t_{0.05,24} = 1.711)$$

Accepted Answer

$$3700 \pm 1.711 	imes \frac{150}{\sqrt{25}}$$

Question 11

Which of the following best describes a Type I error in hypothesis testing?

Accepted Answer

Rejecting a true null hypothesis

Question 12

A researcher performs a two-tailed hypothesis test and obtains a 95% confidence interval that does not contain the hypothesized mean. What can be concluded at the 0.05 significance level?

Accepted Answer

Reject the null hypothesis

Question 13

Which formula is used to determine the minimum sample size $$n$$ required to estimate a population mean with margin of error $$E$$ and known standard deviation $$\sigma at $$confidence level $$1-\alpha$$?

Accepted Answer

$$n = \left( \frac{z_{\alpha/2} \sigma}{E} \$$right)^2$$

Question 14

A student calculates a test statistic $$t = 2.5$$ with $$df = 10$$ for a two-tailed test at $$\alpha = 0.05. $$The critical value is $$t_{0.025,10} = 2.228. $$What is the correct decision?

Accepted Answer

Reject the null hypothesis

Question 15

Which of the following will increase the minimum sample size required for a given margin of error in estimating a mean?

Accepted Answer

Increasing the population standard deviation

Question 16

A financial aid officer wants to estimate the mean student loan debt for graduating seniors to within $$\500$ with 95% confidence. If the standard deviation is $$\$$5400$$, what is the minimum sample size required? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$n = \left( \frac{1.96 	imes 5400}{500} \$$right)^2$$ = 223.24$$, so $$n = 224$$

Question 17

Which of the following is the correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

We are 95% confident that the interval contains the true population mean

Question 18

A researcher is testing $$H_0$$: \mu = 50$$ vs H_a$$: \mu 
eq 50$$ using a sample of n = 16$, sample mean $$\bar{x} = 54$$, and sample standard deviation s = 8$. What is the test statistic?

Accepted Answer

$$t = \frac{54 - 50}{8/\sqrt{16}} = 2.0$$

Question 19

Which of the following is NOT a possible conclusion from a hypothesis test?

Accepted Answer

Accept the alternative hypothesis as absolutely true

Comprehensive Study Notes for Hypothesis Testing, Confidence Intervals, and Probability in Statistics

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