EXAMPLE 4.26

Texting while driving. Texting while driving can be dangerous, but young people want to remain connected. Suppose that 26% of teen drivers text while driving. If we take a sample of 500 teen drivers, what percent would we expect to say that they text while driving?¹¹

The proportion p = 0.26 is a number that describes the population of teen drivers. The proportion of the sample who say that they text while driving is used to estimate p. The proportion is a random variable because repeating the SRS would give a different sample of 500 teen drivers and a different value of .

We will see in the next chapter that in this setting, with teen drivers answering honestly, has approximately the N(0.26, 0.0196) distribution. The mean 0.26 of this distribution is the same as the population proportion because is an unbiased estimate of p. The standard deviation is controlled mainly by the size of the sample.

What is the probability that the survey result differs from the truth about the population by no more than 3 percentage points? We can use what we learned about Normal distribution calculations to answer this question. Because p = 0.26, the survey misses by no more than 3 percentage points if the sample proportion is between 0.23 and 0.29.

Figure 4.11 shows this probability as an area under a Normal density curve. You can find it by software or by standardizing and using Table A. From Table A,

About 87% of the time, the sample will be within 3 percentage points of the proportion p.