Q: A sample of 40 adults is used to estimate the percentage who own a cell phone. If the sample proportion is 0.85, what is the margin of error for a 90% confidence interval? (Assume $$z_{\alpha/2} = 1.645)$$

$$E = 1.645 \sqrt{\dfrac{0.85 \times 0.15}{40}}$$

Question 1

Which of the following best describes a point estimate in statistics?

Accepted Answer

A single value calculated from sample data used to estimate a population parameter.

Question 2

A hospital administrator wants to estimate the proportion of patients who are satisfied with their care. She surveys 200 patients and finds that 160 are satisfied. Construct a 95% confidence interval for the population proportion. Which formula should she use for the margin of error?

Accepted Answer

$$E = z_{\alpha/2} \sqrt{\dfrac{p(1-p)}{n}}$$

Question 3

Which statement about confidence intervals is correct?

Accepted Answer

A confidence interval provides a range of values that is likely to contain the population parameter.

Question 4

A researcher wants to estimate the mean SAT score of students at a university. She takes a random sample of 35 students, finds a sample mean of 1050, and a sample standard deviation of 120. The population standard deviation is unknown. Which distribution should she use to construct a 99% confidence interval for the mean?

Accepted Answer

Student t-distribution

Question 5

What is the critical value $$z_{\alpha/2}$$ for a 95% confidence level?

Accepted Answer

$$z_{\alpha/2} = 1.96$$

Question 6

A sample of 25 nicotine measurements from menthol cigarettes has a standard deviation of 0.18 mg. What are the degrees of freedom for constructing a confidence interval for the population standard deviation?

Accepted Answer

$$df = 24$$

Question 7

When estimating a population mean, under which condition should you use the Student t-distribution instead of the standard normal (z) distribution?

Accepted Answer

When the population standard deviation is unknown and the sample size is small ($$n \leq 30)$$

Question 8

A manufacturer wants to estimate the mean lifetime of a new type of battery. The population standard deviation is known to be 8 hours. What is the minimum sample size required to estimate the mean within 2 hours with 95% confidence?

Accepted Answer

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

Question 9

Which of the following is true about the chi-square distribution?

Accepted Answer

It only takes non-negative values and becomes more symmetric as degrees of freedom increase.

Question 10

A study finds that 70 out of 100 randomly selected medical malpractice lawsuits were dropped or dismissed. What is the point estimate for the proportion of lawsuits dropped or dismissed?

Accepted Answer

$$\hat{p} = 0.7$$

Question 11

If you do not know the population proportion $$p$$, what value should you use for $$p(1-p)$$ when determining the minimum sample size for a proportion estimate?

Accepted Answer

$$p(1-p) = 0.25$$

Question 12

Which formula gives the confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

$$\bar{x} \pm t_{\alpha/2} \dfrac{s}{\sqrt{n}}$$

Question 13

A sample of 40 adults is used to estimate the percentage who own a cell phone. If the sample proportion is 0.85, what is the margin of error for a 90% confidence interval? (Assume $$z_{\alpha/2} = 1.645)$$

Accepted Answer

$$E = 1.645 \sqrt{\dfrac{0.85 	imes 0.15}{40}}$$

Question 14

Which of the following is NOT a requirement for constructing a confidence interval for the population variance using the chi-square distribution?

Accepted Answer

The sample size must be greater than 30.

Question 15

A sample of 20 sitting heights (in mm) of adult women is collected. The sample standard deviation is 35 mm. What is the formula for the 95% confidence interval for the population standard deviation?

Accepted Answer

$$\left( \sqrt{\dfrac{(n-1)s^2$}{\$$chi^2$$_{R}}}, \sqrt{\dfrac{(n-1)s^2$}{\$$chi^2$$_{L}}} \right)$$

Question 16

A scientist wants to be 99% confident that her sample mean is within 5 units of the population mean. The population standard deviation is 20 units. What is the minimum sample size required?

Accepted Answer

$$n = \left( \dfrac{z_{\alpha/2} \cdot 20}{5} \$$right)^2$$

Question 17

Which of the following best describes the confidence level in the context of confidence intervals?

Accepted Answer

The probability that the confidence interval contains the population parameter.

Question 18

A sample of 50 healthy adults has a mean body temperature of 98.2°F and a standard deviation of 0.6°F. Construct a 95% confidence interval for the mean body temperature. Which formula should be used for the margin of error?

Accepted Answer

$$E = t_{\alpha/2} \dfrac{s}{\sqrt{n}}$$

Estimating Parameters and Determining Sample Sizes: Study Notes

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