Q: A biomedical researcher finds the sample mean resting heart rate of 20 adults is 85.5 bpm, with population standard deviation 19.3 bpm. What is the 99% confidence interval for the true mean resting heart rate?

$$[85.5 - 2.57 \times \frac{19.3}{\sqrt{20}}, 85.5 + 2.57 \times \frac{19.3}{\sqrt{20}}]$$

Question 1

Which of the following best describes the goal of statistical inference?

Accepted Answer

To draw conclusions or make propositions about an unknown parameter $$	heta$$ using a random sample from the population.

Question 2

Given a sample $$X_1$$, $$X_2$$, \ldots, X_n$$, which formula correctly defines the sample mean $$\bar{X}$$?

Accepted Answer

$$\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$$

Question 3

A pharmaceutical company collects recovery times (in days) for 50 patients after a new treatment. The sample mean is 12.4 days and the population standard deviation is 5.6 days. What is the standard error of the sample mean?

Accepted Answer

$$SE = \frac{5.6}{\sqrt{50}}$$

Question 4

Which statement about a 95% confidence interval for the population mean is correct?

Accepted Answer

It means that if we repeatedly take samples and construct intervals, about 95% of those intervals will contain the population mean.

Question 5

Given a sample $$X_1$$, $$X_2$$, \ldots, X_n$$, which formula gives the sample variance S^2$?

Accepted Answer

$$S^2$$ = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X}$$)^2$$

Question 6

A researcher records the weights (in grams) of 10 rats: 27.7, 31.5, 30.9, 29.6, 27.0, 38.1, 32.4, 31.1, 36.7, 28.3. What is the sample proportion of rats weighing more than 35 grams?

Accepted Answer

$$\hat{p} = \frac{2}{10}$$

Question 7

Which of the following is the correct formula for a $$(1-\alpha)$$ confidence interval for the mean $$\mu of a $$normal population with known variance $$\$$sigma^2$$ and sample mean $$\bar{X}$$?

Accepted Answer

$$\left[ \bar{X} - z_{\alpha/2} \frac{\sigma}{\sqrt{n}}, \bar{X} + z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \right]$$

Question 8

A biomedical researcher finds the sample mean resting heart rate of 20 adults is 85.5 bpm, with population standard deviation 19.3 bpm. What is the 99% confidence interval for the true mean resting heart rate?

Accepted Answer

$$[85.5 - 2.57 	imes \frac{19.3}{\sqrt{20}}, 85.5 + 2.57 	imes \frac{19.3}{\sqrt{20}}]$$

Question 9

Which of the following statements about the sampling distribution of the sample mean $$\bar{X} is $$correct when $$X \sim N(\mu, \$$sigma^2$$)$$?

Accepted Answer

$$\bar{X} is $$normally distributed with mean $$\mu$$ and variance $$\$$sigma^2$$/n$$

Question 10

A researcher compares cholesterol levels between two diets. Diet A: $$n=40$$, $$\bar{X}_A=195$$, $$\sigma_A=20. $$Diet B: $$n=35$$, $$\bar{X}_B=210$$, $$\sigma_B=25. $$What is the standard error for the difference in sample means?

Accepted Answer

$$SE = \sqrt{\frac{20^2$}{40} + \frac{25^2$}{35}}$$

Question 11

Which of the following is NOT a common statistical inference problem?

Accepted Answer

Data collection

Question 12

If a 90% confidence interval is constructed from 20 random samples, how many intervals are likely to contain the population mean?

Accepted Answer

18

Question 13

Which formula gives the sample proportion $$\hat{p} of $$units with a characteristic in a sample of size $$n$$?

Accepted Answer

$$\hat{p} = \frac{y}{n}$$, where $$y is $$the number of units with the characteristic

Question 14

A wildlife biologist samples 16 animals, with $$\sum x_i = 49$$ and $$\sum x_i^2$ = 374. $$What is the sample mean?

Accepted Answer

$$\bar{x} = \frac{49}{16}$$

Question 15

Which of the following best describes the margin of error in a confidence interval?

Accepted Answer

It is the quantity added and subtracted from the point estimate to form the confidence interval.

Question 16

For a normal population with known variance, which value is used to determine the width of the confidence interval for the mean?

Accepted Answer

$$z_{\alpha/2}$$

Question 17

Which statement is true about sample statistics?

Accepted Answer

Sample statistics are random variables that summarize sample data.

Question 18

A $$(1-\alpha)$$ confidence interval for $$	heta is $$constructed as $$[\hat{	heta} - m, \hat{	heta} + m]. $$What does $$(1-\alpha)$$ represent?

Accepted Answer

The confidence level, or the proportion of intervals expected to contain $$	heta.$$

Statistical Inference: Point and Interval Estimation

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