# AP Statistics : Confidence Intervals and Regression

## Example Questions

### Example Question #1 : Confidence Intervals And Regression

You estimate a regression model with and , where is the beta coefficient and is the standard error.  Construct 95% confidence intervals for .    Explanation:

To construct 95% confidence intervals for , we simply take the coefficient and add/subtract . This is because is assumed to follow a symmetrical distribution (the normal), and 95% of the values in the sampling distribution are contained within 1.96 standard errors of .

### Example Question #1 : How To Find Confidence Intervals

Which of the following statements are correct about confidence intervals?

The width of a confidence interval does not change as the sample size increases and increases as the confidence level increases.

The width of a confidence interval increases as the sample size increases and increases as the confidence level increases.

The width of a confidence interval decreases as the sample size increases and increases as the confidence level increases.

The width of a confidence interval decreases as the sample size increases and increases as the confidence level decreases.

The width of a confidence interval increases as the sample size increases and increases as the confidence level decreases.

The width of a confidence interval decreases as the sample size increases and increases as the confidence level increases.

Explanation:

Larger samples give narrower intervals. We are able to estimate a population proportion more precisely with a larger sample size.

As the confidence level increases the width of the confidence interval also increases. A larger confidence level increases the chance that the correct value will be found in the confidence interval. This means that the interval is larger.

### Example Question #1 : Confidence Intervals And Regression

You are asked to create a confidence interval with a margin of error no larger than while sampling from a normally distributed population with a standard deviation of . What is the minimum required sample size?      Explanation:

Keep in mind that the margin of error for a confidence interval based on a normal population is equal to , where is the -score corresponding to the desired confidence level.

From the problem, we can tell that and . We can then solve for algebraically:    The minimum sample size is rounded up, which is . If you are unsure on problems like these, you can check the margin of error for your answer rounded down and then rounded up (in this case, for and .)

### Example Question #1 : How To Find Confidence Intervals

Jim calculated a confidence interval for the mean height in inches of boys in his high school. He is not sure how to interpret this interval. Which of the following explains the meaning of confidence. of all possible sample means fall within Jim's interval.

In the long run, of all confidence intervals calculated from the same population will contain the true mean height.

There is a chance that Jim's interval contains the true mean height. of boys' heights fall with the interval Jim calculated.

In the long run, of all confidence intervals calculated from the same population will contain the true mean height.

Explanation:

95% confidence means that the methods Jim uses to calculate his confidence interval give him correct results 95% of the time. It does not mean that there is a 95% chance that the true mean will be inside the interval. It also does not mean that 95% of all heights or possible sample means fall within the interval.

### Example Question #1 : How To Find Confidence Intervals

The number of hamburgers served by McGregors per day is normally distributed and has a mean of hamburgers and a standard deviation of Find the range of customers served on the middle percent of days.      Explanation:

First, find the first quartile of the distribution. Then, find the third quartile of the distribution. ### All AP Statistics Resources 