Example 2

Surveys According to a national survey, 28% of adults in the United States prefer to order American food when dining out. You are conducting a random survey of 200 adults. What is the probability that at most 50 of them will prefer to order American food when dining out?

This is a binomial experiment with n = 200 n equals 200  and p = 0.28 . p equals , 0.28 , .  The binomial distribution shows the number x of adults who prefer to order American food when dining out.

• Step 1 Check to see if you can use a normal distribution to approximate the binomial distribution.

  n p = 200 · 0.28 = 56 ; 56 > 5 n p equals 200 middle dot , 0.28 , equals 56 semicolon 56 greater than 5

  n ( 1 − p ) = 200 · 0.72 = 144 ; 144 > 5 . n open 1 minus p close equals 200 middle dot , 0.72 , equals 144 semicolon 144 greater than 5 .

  You can approximate the binomial distribution with a normal distribution.

• Step 2 Find the mean and standard deviation for the binomial standard deviation.

  mean  μ = n p = 56 mean , mu equals n p equals 56

  standard deviation  σ = n p ( 1 − p ) = 200 ( 0.28 ) ( 0.72 ) ≈ 6 standarddeviation . sigma equals . square root of n p open 1 minus p close end root . equals . square root of 200 open , 0.28 , close open , 0.72 , close end root . almost equal to 6

• Step 3 Use the standard normal curve.

• The z-score for 50 is z = X − μ σ = 50 − 56 6 = − 1 , z equals . fraction x minus mu , over sigma end fraction . equals . fraction 50 minus 56 , over 6 end fraction . equals negative 1 . comma  or 1 standard deviation below the mean.

  P ( x ≤ 50 ) = 0.5 − 0.34 = 0.16 p open x less than or equal to 50 close equals 0.5 minus , 0.34 , equals , 0.16

  The probability that at most 50 of the people surveyed will say that they prefer to order American food when dining out is about 16%.

  

Exercises

1. Manufacturing A company manufactures batteries having lifespans that are normally distributed, with a mean of 45 mo and a standard deviation of 5 mo. Find the probability that a battery chosen at random will have a lifespan of 50–55 mo.
2. Communications According to a national survey, 16% of U.S. adults use the Internet to make telephone calls. In a random survey of 100 adults, what is the probability that at most 20 of them use the Internet to make telephone calls? (Source: The Harris Poll #51, June 7, 2007. Copyright © 2007 Harris Interactive, Inc.)
3. Pets According to a national survey, 92% of U.S. dog owners consider their pets to be members of the family. In a random survey of 150 dog owners, what is the probability that at most 132 of them consider their pets to be members of the family? (Source: The Harris Poll ^® #120, December 4, 2007. Copyright © 2007 Harris Interactive, Inc.)

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