Q: A sample of 16 batteries has a mean lifetime of 500 hours and a standard deviation of 40 hours. Construct a 90% confidence interval for the population mean battery lifetime. The critical value from the t-table for 15 degrees of freedom at $$ \alpha/2 = 0.05 is 1.753. $$What is the margin of error?

$$ E = 1.753 \times \dfrac{40}{\sqrt{16}} = 17.53 $$

Question 1

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

Accepted Answer

A value calculated from a sample that is used to estimate a population parameter.

Question 2

A researcher wants to estimate the proportion of defective items produced by a factory. She takes a random sample of 200 items and finds that 18 are defective. Construct a 95% confidence interval for the population proportion of defective items. Which formula should she use for the margin of error?

Accepted Answer

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

Question 3

Suppose you want to estimate the mean systolic blood pressure of adults in a city. You collect a simple random sample of 25 adults, and the sample mean is 122 mmHg with a sample standard deviation of 15 mmHg. Which distribution should you use to construct a 99% confidence interval for the population mean?

Accepted Answer

Student's t-distribution with 24 degrees of freedom

Question 4

A sample of 16 batteries has a mean lifetime of 500 hours and a standard deviation of 40 hours. Construct a 90% confidence interval for the population mean battery lifetime. The critical value from the t-table for 15 degrees of freedom at $$ \alpha/2 = 0.05  is 1.753. $$What is the margin of error?

Accepted Answer

$$ E = 1.753 	imes \dfrac{40}{\sqrt{16}} = 17.53 $$

Question 5

Which of the following statements about the level of confidence is correct?

Accepted Answer

It represents the expected proportion of intervals that will contain the parameter if a large number of samples are taken.

Question 6

A hospital administrator wants to estimate the variance in patient wait times. She collects a random sample of 10 wait times and calculates a sample variance of 25 minutes$$^2. $$Which distribution should she use to construct a confidence interval for the population variance?

Accepted Answer

Chi-square distribution with 9 degrees of freedom

Question 7

If a 95% confidence interval for a population mean is calculated as (45.5, 54.5), what is the correct interpretation?

Accepted Answer

We are 95% confident that the true population mean lies between 45.5 and 54.5.

Question 8

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

Accepted Answer

It is used when the population standard deviation is known.

Question 9

A quality control manager wants to determine the minimum sample size needed to estimate the proportion of defective products within 2% of the true value with 95% confidence. Which formula should she use?

Accepted Answer

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

Question 10

Which of the following best describes the role of $$ \alpha  ($$alpha) in constructing confidence intervals?

Accepted Answer

It represents the probability of making a Type I error or the chance of error we are willing to take.

Question 11

A researcher is constructing a confidence interval for the standard deviation of a population. Which of the following is the correct form of the confidence interval for the population variance $$ \$$sigma^2$$ $$ using the chi-square distribution?

Accepted Answer

$$ \left( \dfrac{(n-1)s^2$}{\$$chi^2$$_{\alpha/2}}, \dfrac{(n-1)s^2$}{\$$chi^2$$_{1-\alpha/2}} \right) $$

Question 12

A sample of 30 students has a mean test score of 78 and a standard deviation of 10. What is the 95% confidence interval for the population mean if $$ t_{0.025, 29} = 2.045 $$? (Round your answer to one decimal place.)

Accepted Answer

($$ 74.3 $$, $$ 81.7 )$$

Question 13

Which of the following scenarios requires the use of the chi-square distribution to construct a confidence interval?

Accepted Answer

Estimating the population variance of the time it takes for a chemical reaction to complete.

Question 14

A sample of 50 light bulbs has a sample mean lifetime of 1200 hours and a sample standard deviation of 100 hours. The population is known to be normally distributed. What is the 99% confidence interval for the mean lifetime? ($$ t_{0.005, 49} = 2.678 )$$

Accepted Answer

($$ 1162.1 $$, $$ 1237.9 )$$

Question 15

Which of the following is a correct statement about the relationship between confidence level and margin of error?

Accepted Answer

Increasing the confidence level increases the margin of error, all else being equal.

Question 16

A medical researcher wants to estimate the mean recovery time for a new drug. She collects a sample of 36 patients, finds a mean recovery time of 14 days, and a standard deviation of 3 days. What is the 95% confidence interval for the mean recovery time? ($$ t_{0.025, 35} = 2.030 )$$

Accepted Answer

($$ 12.99 $$, $$ 15.01 )$$

Question 17

Which of the following is a correct interpretation of a 90% confidence interval for a population proportion?

Accepted Answer

If we repeated the sampling process many times, about 90% of the constructed intervals would contain the true population proportion.

Question 18

A sample of 12 patients has a sample standard deviation of 5.2. What is the 95% confidence interval for the population standard deviation? ($$ \$$chi^2$$_{0.025, 11} = 21.92 $$, $$ \$$chi^2$$_{0.975, 11} = 3.82 )$$

Accepted Answer

($$ 3.7 $$, $$ 9.2 )$$

Question 19

Which of the following is a correct formula for the confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

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

Chapter 9: Confidence Intervals for Population Parameters

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