Question 1

Which of the following is the correct formula for a point estimate of the population mean $$\mu$$ based on a sample of $$n$$ observations $$x_1$$, $$x_2$$, ..., x_n$$?

Accepted Answer

$$\displaystyle \frac{\sum_{i=1}^n x_i}{n}$$

Question 2

A researcher wants to estimate the average miles per gallon (mpg) for a new car model. She collects a random sample of 16 cars and calculates a sample mean of 43.2 mpg with a sample standard deviation of 2.5 mpg. Which distribution should she use to construct a confidence interval for the population mean?

Accepted Answer

Student's $$t-$$distribution with 15 degrees of freedom

Question 3

Which of the following is NOT a property of Student's $$t-$$distribution?

Accepted Answer

It is always used when the population standard deviation is known

Question 4

Suppose a sample of size $$n=10 is $$drawn from a normal population with unknown mean and unknown standard deviation. What is the distribution of the standardized sample mean?

Accepted Answer

$$t-$$distribution with 9 degrees of freedom

Question 5

Which of the following is the correct formula for the margin of error $$E$$ when constructing a confidence interval for a population mean using Student's $$t-$$distribution?

Accepted Answer

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

Question 6

A 95% confidence interval for a population mean is calculated as $$\bar{x} \pm t_{0.025} \cdot \dfrac{s}{\sqrt{n}}. $$What does the value $$t_{0.025}$$ represent?

Accepted Answer

The critical value from the $$t-$$distribution with $$n-1$$ degrees of freedom that leaves 2.5% in the upper tail

Question 7

Which of the following is NOT a required condition for constructing a confidence interval for a population mean using the $$t-$$distribution?

Accepted Answer

The population standard deviation is known

Question 8

Given a sample mean $$\bar{x} = 42.5$$, sample standard deviation $$s = 2.1$$, sample size $$n = 16$$, and $$t_{0.025, 15} = 2.131$$, what is the 95% confidence interval for the population mean $$\mu$$?

Accepted Answer

$$42.5 \pm 2.131 \cdot \dfrac{2.1}{4}$$

Question 9

If the sample size increases while the sample standard deviation remains constant, what happens to the width of the confidence interval for the population mean?

Accepted Answer

It decreases

Question 10

Which formula gives the required sample size $$n to $$estimate a population mean $$\mu$$ with margin of error $$E$$ and confidence level $$(1-\alpha)$$, assuming the population standard deviation is unknown and estimated by $$s$$?

Accepted Answer

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

Question 11

A medical researcher wants to estimate the average recovery time for a new drug. She collects a random sample of 25 patients and finds a sample mean of 8.2 days and a sample standard deviation of 1.5 days. What is the margin of error for a 95% confidence interval if $$t_{0.025, 24} = 2.064$$?

Accepted Answer

$$2.064 \cdot \dfrac{1.5}{5}$$

Question 12

Why is the $$t-$$distribution used instead of the standard normal distribution when constructing a confidence interval for a population mean with unknown standard deviation and small sample size?

Accepted Answer

Because the $$t-$$distribution accounts for the additional uncertainty in estimating the population standard deviation from the sample

Question 13

A confidence interval for the mean miles per gallon of a car is calculated as (41.2, 44.8). What does this interval mean?

Accepted Answer

We are 95% confident that the true mean miles per gallon for the population lies between 41.2 and 44.8

Question 14

Which of the following statements about the robustness of the $$t-$$interval procedure is TRUE?

Accepted Answer

The $$t-$$interval procedure is robust to moderate departures from normality, especially for large sample sizes

Question 15

A researcher wants to estimate the mean blood pressure in a population. She wants the margin of error to be no more than 2 mmHg with 95% confidence. If the estimated standard deviation is 8 mmHg and $$t_{0.025, n-1} \approx 2$$, what is the minimum sample size required?

Accepted Answer

$$n = \left( \dfrac{2 \cdot 8}{2} \$$right)^2$$ = 16$$

Question 16

If the sample mean and sample standard deviation are not robust to outliers, which method can be used to construct a confidence interval for the mean?

Accepted Answer

Bootstrapping

Question 17

Which of the following best describes the effect of using a larger critical value (e.g., $$t_{\alpha/2}) $$when constructing a confidence interval for the mean?

Accepted Answer

The confidence interval becomes wider

Question 18

A sample of 20 students' test scores has a mean of 78 and a standard deviation of 6. What is the 99% confidence interval for the population mean if $$t_{0.005, 19} = 2.861$$?

Accepted Answer

$$78 \pm 2.861 \cdot \dfrac{6}{\sqrt{20}}$$

9.2 Estimating a Population Mean and Confidence Intervals

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