Q: A sample of 15 patients has a mean cholesterol level of 210 mg/dL and a standard deviation of 30 mg/dL. What is the 95% confidence interval for the population mean? (Use $$t^* = 2.145$$ for $$df = 14.)$$

$$210 \pm 2.145 \times \frac{30}{\sqrt{15}}$$

Question 1

A researcher wants to estimate the mean systolic blood pressure of adults in a city. She collects a sample of 36 adults and finds a sample mean of 130 mmHg with a sample standard deviation of 12 mmHg. Construct a 95% confidence interval for the population mean. Which formula should she use?

Accepted Answer

$$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$$

Question 2

Which of the following best describes the meaning of a z-score of $$-2.0$$ for a data value in a normal distribution?

Accepted Answer

The value is 2 standard deviations below the mean.

Question 3

A sample of 25 students has a mean test score of 78 and a standard deviation of 10. What is the standard error of the mean?

Accepted Answer

$$\frac{10}{\sqrt{25}}$$

Question 4

Which formula would you use to calculate the required sample size to estimate a population mean with a specified margin of error $$E$$ and confidence level $$z^*$$?

Accepted Answer

$$n = \left(\frac{z^* \sigma}{E}\$$right)^2$$

Question 5

A pharmaceutical company claims that their new drug is effective in 80% of cases. In a clinical trial with 100 patients, 72 showed improvement. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$0.72$$

Question 6

Which formula is used to calculate the z-score for an individual value $$x in a $$population with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 7

A 99% confidence interval for a population mean is wider than a 95% confidence interval because:

Accepted Answer

The critical value $$z^* is $$larger for 99% confidence.

Question 8

Which formula is used to calculate the margin of error for a confidence interval for a population proportion?

Accepted Answer

$$z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 9

A sample of 15 patients has a mean cholesterol level of 210 mg/dL and a standard deviation of 30 mg/dL. What is the 95% confidence interval for the population mean? (Use $$t^* = 2.145$$ for $$df = 14.)$$

Accepted Answer

$$210 \pm 2.145 	imes \frac{30}{\sqrt{15}}$$

Question 10

Which of the following is the correct formula for the test statistic when testing a population mean with a known population standard deviation?

Accepted Answer

$$z = \frac{\bar{x} - \mu_0$}{\sigma / \sqrt{n}}$$

Question 11

A quality control manager wants to test if the proportion of defective items in a shipment is less than 5%. Which hypothesis test should be used?

Accepted Answer

One-tailed test for a population proportion

Question 12

If a sample of size $$n = 50$$ has a sample variance $$s^2 = 25$, what is the standard error of the sample mean?

Accepted Answer

$$\frac{5}{\sqrt{50}}$$

Question 13

Which formula is used to calculate the binomial probability of exactly $$k$$ successes in $$n$$ trials with probability $$p of $$success?

Accepted Answer

$$P(X = k) = \frac{n!}{k!(n-k)!} p^k (1-p$$)^{n-k}$$

Question 14

A researcher calculates a Pearson correlation coefficient $$r = 0.85$$ for two variables. Which formula should be used to test the significance of this correlation?

Accepted Answer

$$t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2$}}$$

Question 15

Which of the following is the correct formula for the standard deviation of a binomial distribution with parameters $$n$$ and $$p$$?

Accepted Answer

$$\sqrt{np(1-p)}$$

Question 16

A sample of 20 patients is used to estimate the mean recovery time from a new treatment. Which distribution should be used to construct a confidence interval for the mean if the population standard deviation is unknown?

Accepted Answer

t-distribution

Question 17

Which formula is used to calculate the test statistic for a sample variance when testing against a hypothesized population variance $$\$$sigma_0$$^2$$?

Accepted Answer

$$\$$chi^2$$ = \frac{(n-1)s^2$}{\$$sigma_0$$^2}$$

Question 18

A 90% confidence interval for a population proportion is calculated as $$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. $$What is the value of $$z^*$$ for a 90% confidence level?

Accepted Answer

$$1.645$$

Question 19

Which formula is used to calculate the sample mean $$\bar{x}$$ from a set of $$n$$ observations $$x_1$$, $$x_2$$, ..., x_n$$?

Accepted Answer

$$\bar{x} = \frac{\sum x_i}{n}$$

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