7.3 Using the central limit theorem  (Page 5/31)

For part a, you include 150 so P ( X ≥ 150) has normal approximation P ( Y ≥ 149.5) = 0.8641.

normalcdf (149.5,10^99,159,8.6447) = 0.8641.

For part b, you include 160 so P ( X ≤ 160) has normal appraximation P ( Y ≤ 160.5) = 0.5689.

normalcdf (0,160.5,159,8.6447) = 0.5689

For part c, you exclude 155 so P ( X >155) has normal approximation P ( y >155.5) = 0.6572.

normalcdf (155.5,10^99,159,8.6447) = 0.6572.

For part d, you exclude 147 so P ( X <147) has normal approximation P ( Y <146.5) = 0.0741.

normalcdf (0,146.5,159,8.6447) = 0.0741

For part e, P ( X = 175) has normal approximation P (174.5< Y <175.5) = 0.0083.

normalcdf (174.5,175.5,159,8.6447) = 0.0083

Because of calculators and computer software that let you calculate binomial probabilities for large values of n easily, it is not necessary to use the the normal approximation to the binomial distribution, provided that you have access to these technology tools. Most school labs have Microsoft Excel, an example of computer software that calculates binomial probabilities. Many students have access to the TI-83 or 84 series calculators, and they easily calculate probabilities for the binomial distribution. If you type in "binomial probability distribution calculation" in an Internet browser, you can find at least one online calculator for the binomial.

For [link] , the probabilities are calculated using the following binomial distribution: ( n = 300 and p = 0.53). Compare the binomial and normal distribution answers. See Discrete Random Variables for help with calculator instructions for the binomial.

P ( X ≥ 150) : 1 - binomialcdf (300,0.53,149) = 0.8641

P ( X ≤ 160) : binomialcdf (300,0.53,160) = 0.5684

P ( X >155) : 1 - binomialcdf (300,0.53,155) = 0.6576

P ( X <147) : binomialcdf (300,0.53,146) = 0.0742

P ( X = 175) :(You use the binomial pdf.) binomialpdf (300,0.53,175) = 0.0083

References

Data from the Wall Street Journal.

“National Health and Nutrition Examination Survey.” Center for Disease Control and Prevention. Available online at http://www.cdc.gov/nchs/nhanes.htm (accessed May 17, 2013).

Chapter review

The central limit theorem can be used to illustrate the law of large numbers. The law of large numbers states that the larger the sample size you take from a population, the closer the sample mean $\overline{x}$ gets to μ .

Use the following information to answer the next ten exercises: A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely so the distribution of weights is uniform. A sample of 100 weights is taken.

Use the following information to answer the next five exercises: The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.

Use the following information to answer the next eight exercises: A uniform distribution has a minimum of six and a maximum of ten. A sample of 50 is taken.