Confidence Intervals

Overview

Smartphones are gaining teenage users. Some 23% of all those ages 12-17 say they have a smartphone and ownership is highest among older teens: 31% of those ages 14-17 have a smartphone, compared with just 8% of youth ages 12-13.

Teens, Smartphones & TextingTexting Volume Is Up while the Frequency of Voice Calling Is Down

http://pewresearch.org/pubs/2223/teens-cellphones-texting-phone-calls?src=prc=newsletter

Recall Sampling Distributions

The statistics quoted by PEW Research are based on a sample. Let’s say that exactly 30% of all teens age 14 to 17 in the U.S. have a smartphone.

If we were to take a random sample of 40 teens age 14 to 17 and record the proportion that had a smartphone, what proportions would be reasonably likely in our sample? In other words, if we did many such samples, the middle 95% of sample proportions would be between _____ and ______.

Recall Sampling Distributions

A normal model is appropriate, because . . .

So the middle 95% of sample proportions should fall within 1.96 standard deviations of 0.30.

So or 0.158 and 0.442.

Suppose that in a random sample of 40 retired women, 45% of the women travel more than they did while they were working. Find the 95% confidence interval for the proportion of all retired women who travel more.

Capture Rate

Here is a set of 100 confidence intervals created from 100 random samples of size 40. p = 0.45.

Source: 2010 AP Exam

Unfortunately, we wouldn’t know the standard deviation is 107. We’d have to estimate the standard deviation using the sample standard deviation.

We adjust for this by changing the critical value. We don’t usethe cutoffs for the normal distribution (z), we use an adjustedvalue called t.

t is larger than z, so it make the intervals wider, and therefore more likely to capture the population mean. In this case we use t = 3.182 for 3 degrees of freedom. (df = n – 1)

Source: 2002 AP Exam

Answer: A

Source: 2002 AP Exam

Answer: E

Source: 2002 AP Exam

Answer: D

Source: 1997 AP Exam

1. USA Today reported that speed skater Bonnie Blake had “won the USA’s heart,” according to a USA Today/CNN/Gallup poll conducted on the final Thursday of the 1994 Winter Olympics. When asked who was the hero of the Olympics, 65 percent of the respondents chose Blair, who won five gold medals. The poll of 615 adults, done by telephone, had a margin of error of 4 percent. Which of the following statements best describes what is meant by the 4 percent margin of error?

(A) About 4 percent of adults were expected to change their minds between the time of the poll and its publication in USA Today. (B) About 4 percent of adults did not have telephones. (C) About 4 percent of the 615 adults polled refused to answer. (D) Not all of the 615 adults knew anything about the Olympics. (E) The difference between the sample percentage and the population percentage is likely to be less than 4 percent.

Answer: E

Source: 2009 AP Exam

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