Q: If a sample mean is $$ \bar{x} = 380.67 $$, the sample standard deviation is $$ s = 42.47 $$, the sample size is $$ n = 10 $$, and the critical value $$ t_{0.025} = 2.262 $$, what is the lower bound of the 95% confidence interval for the population mean?

$$ 380.67 - 2.262 \times \frac{42.47}{\sqrt{10}} = 350.29 $$

Question 1

Which of the following best describes the difference between a population parameter and a sample statistic?

Accepted Answer

A population parameter is a fixed value that describes a characteristic of the entire population, while a sample statistic is a value calculated from a sample and used to estimate the parameter.

Question 2

A researcher wants to estimate the average number of hours college students sleep per night. She collects data from a random sample of 100 students. Which confidence interval formula should she use?

Accepted Answer

Use the confidence interval for a population mean: $$ \bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}} $$

Question 3

Which of the following is NOT a requirement for constructing a confidence interval for a population proportion?

Accepted Answer

The sample size must be at least 30.

Question 4

A boxplot of a dataset shows no outliers and a normal probability plot yields a p-value of 0.87. What does this indicate about the data?

Accepted Answer

The data are approximately normal and suitable for constructing a confidence interval for the mean.

Question 5

Which of the following best describes the difference between an observational study and a randomized controlled experiment?

Accepted Answer

A randomized controlled experiment involves random assignment of treatments to subjects, while an observational study does not manipulate treatments.

Question 6

Suppose you want to estimate the proportion of adults in a city who support a new public policy. You survey 400 randomly selected adults and find that 220 support the policy. Which formula should you use to construct a 95% confidence interval for the population proportion?

Accepted Answer

$$ \hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$

Question 7

Which of the following is a measure of dispersion?

Accepted Answer

Standard deviation

Question 8

A developmental mathematics instructor wants to estimate the typical amount of time students dedicate to studying mathematics in a week. She collects data from 50 students. Which confidence interval should she construct?

Accepted Answer

Confidence interval for a population mean

Question 9

If a sample mean is $$ \bar{x} = 380.67 $$, the sample standard deviation is $$ s = 42.47 $$, the sample size is $$ n = 10 $$, and the critical value $$ t_{0.025} = 2.262 $$, what is the lower bound of the 95% confidence interval for the population mean?

Accepted Answer

$$ 380.67 - 2.262 	imes \frac{42.47}{\sqrt{10}} = 350.29 $$

Question 10

Which of the following statements about the Central Limit Theorem is correct?

Accepted Answer

As the sample size increases, the distribution of the sample mean becomes approximately normal, regardless of the population's distribution.

Question 11

Which of the following is a correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

We are 95% confident that the interval calculated from this sample contains the population mean.

Question 12

Which of the following is NOT a graphical summary of data?

Accepted Answer

Standard deviation

Question 13

A researcher wants to know if chewing food longer reduces caloric intake at dinner. She collects calorie consumption data from 75 subjects who chew twice as long. Which confidence interval should she construct?

Accepted Answer

Confidence interval for a population mean

Question 14

Which of the following is a requirement for constructing a confidence interval for a population mean when the sample size is less than 30?

Accepted Answer

The data must come from a population that is at least approximately normal with no outliers.

Question 15

Which of the following is the correct formula for the standard deviation of the sampling distribution of the sample proportion $$ \hat{p} $$?

Accepted Answer

$$ \sqrt{\frac{p(1-p)}{n}} $$

Question 16

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

Accepted Answer

The maximum expected difference between the point estimate and the true population parameter.

Question 17

A Gallup poll asks 1016 adults if moral values are getting better or worse. Which confidence interval should be constructed?

Accepted Answer

Confidence interval for a population proportion

Question 18

Which of the following is a correct formula for the lower bound of a confidence interval for a population mean?

Accepted Answer

$$ \bar{x} - t_{\alpha/2} \frac{s}{\sqrt{n}} $$

Review of Confidence Intervals and Basic Statistical Inference

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