Math 140 Confidence Intervals #7

Constructing Confidence Intervals for 1 population mean

and 1 population proportion (percentage)

Confidence intervals give two values that we think the population value is in between. To

construct a confidence interval, we start with the sample value (point estimate) and then add and

subtract a certain number of standard deviations from the sample value. These standard

deviations are also called standard errors. The number of standard errors is the critical z-

scores corresponding to a certain confidence level. Later, we will see that we can also use the t-

distribution to calculate the number of standard errors, but for now we will just use the standard

normal distribution (z-scores).

Directions: For numbers 1-3, use the formula c

sx Z

n and your calculator to calculate

the confidence interval estimate of the population mean .

1. A random sample of 650 high school students has a normal distribution. The sample

mean average ACT exam score was 21 with a 3.2 sample standard deviation. Construct

a 99% confidence interval estimate of the population mean average ACT exam.

2. A random sample of 200 adults found that they had a sample mean temperature of 98.2

degrees and a standard deviation of 1.8 degrees. Construct a 95% confidence interval

estimate of the population mean body temperature of adults. Does the confidence

interval indicate that normal body temperature could be 98.6 degrees?

3. A random sample of 315 adults found that the sample mean amount or credit card debt was

$435 with a standard deviation of $106. Construct a 90% confidence interval estimate of the

population mean amount of credit card debt.

Directions: For numbers 4-6, use the formula ˆ ˆ1

ˆc

p pp Z

n

and your calculator to

calculate the confidence interval estimate of the population percent p. You may have to

use the formula ˆx

pn

to calculate the sample percent p̂ . Also remember to write the

sample proportion p̂ as a decimal before plugging into the formula.

4. In a random sample of 72 adults in Santa Clarita, CA, each person was asked if they

support the death penalty. 31 adults in the sample said that they do support the death

penalty. What was the sample proportion of adults in Santa Clarita that support the death

penalty? Now calculate a 95% confidence interval population estimate of people in Santa

Clarita that support the death penalty. Remember to use the appropriate critical value Z-

score for each.

5. In a random sample of 400 Americans, each person was asked if they are satisfied with

the amount of vacation time they given by their employers. 84% of them said that they

were not satisfied with their vacation time. Calculate the following. What was the

sample proportion of Americans that were not satisfied with their vacation time? Now

construct a 99% confidence interval in order to estimate the percent of Americans that are

not satisfied with their vacation time.

6. What percent of eligible Americans vote? In 2008, a random sample of 3000 American

adults that were eligible to vote was taken and we found that 2040 of them voted.

Construct a 90% confidence interval estimate of the population percent of Americans that

vote. Now construct another confidence interval. This time construct a 90% confidence

interval estimate of the population percent of Americans that do not vote. Hint: For the

“do not vote” group, the sample percent will change.