6 Normal Curves and Sampling Distributions Copyright © Cengage Learning. All rights reserved.

Normal Approximation to Binomial Distribution and Section 6.6 Normal Approximation to Binomial Distribution and to p Distribution ^ Copyright © Cengage Learning. All rights reserved.

Focus Points State the assumptions needed to use the normal approximation to the binomial distribution. Compute  and  for the normal approximation. Use the continuity correction to convert a range of r values to a corresponding range of normal x values. Convert the x values to a range of standardized z scores and find desired probabilities.

Focus Points Describe the sampling distribution for proportions .

Normal Approximation to Binomial Distribution and to p Distribution ^ The probability that a new vaccine will protect adults from cholera is known to be 0.85. The vaccine is administered to 300 adults who must enter an area where the disease is prevalent. What is the probability that more than 280 of these adults will be protected from cholera by the vaccine? This question falls into the category of a binomial experiment, with the number of trials n equal to 300, the probability of success p equal to 0.85, and the number of successes r greater than 280.

Normal Approximation to Binomial Distribution and to p Distribution ^ It is possible to use the formula for the binomial distribution to compute the probability that r is greater than 280. However, this approach would involve a number of tedious and long calculations. There is an easier way to do this problem, for under the conditions stated below, the normal distribution can be used to approximate the binomial distribution.

Normal Approximation to Binomial Distribution and to p Distribution ^

Example 14 – Binomial distribution graphs Graph the binomial distributions for which p = 0.25, q = 0.75, and the number of trials is first n = 3, then n = 10, then n = 25, and finally n = 50. Solution: The authors used a computer program to obtain the binomial distributions for the given values of p, q, and n. The results have been organized and graphed in Figures 6-34, 6-35, 6-36, and 6-37. Figure 6-34

Example 14 – Solution cont’d Figure 6-35 Figure 6-36 Good Normal Approximation; np > 5 and nq > 5 Figure 6-37

Example 14 – Solution cont’d When n = 3, the outline of the histogram does not even begin to take the shape of a normal curve. But when n = 10, 25, or 50, it does begin to take a normal shape, indicated by the red curve. From a theoretical point of view, the histograms in Figures 6-35, 6-36, and 6-37 would have bars for all values of r from r = 0 to r = n. However, in the construction of these histograms, the bars of height less than 0.001 unit have been omitted—that is, in this example, probabilities less than 0.001 have been rounded to 0.

Normal Approximation to Binomial Distribution and to p Distribution ^ Procedure:

Normal Approximation to Binomial Distribution and to p Distribution ^ P(6  r  10) Is Approximately Equal to P(5.5  x  10.5) Figure 6-38

Sampling Distributions for Proportions

Sampling Distributions for Proportions We have studied the sampling distribution for the mean. Now we have the tools to look at sampling distributions for proportions. Suppose we repeat a binomial experiment with n trials again and again and, for each n trials, record the sample proportion of successes

Sampling Distributions for Proportions The values form a sampling distribution for proportions.

Example 16 – Sampling distribution of p ^ The annual crime rate in the Capital Hill neighborhood of Denver is 111 victims per 1000 residents. This means that 111 out of 1000 residents have been the victim of at least one crime. These crimes range from relatively minor crimes (stolen hubcaps or purse snatching) to major crimes (murder).

Example 16 – Sampling distribution of p ^ cont’d The Arms is an apartment building in this neighborhood that has 50 year round residents. Suppose we view each of the n = 50 residents as a binomial trial. The random variable r (which takes on values 0, 1, 2, . . . , 50) represents the number of victims of at least one crime in the next year.

Example 16(a) – Sampling distribution of p ^ cont’d What is the population probability p that a resident in the Capital Hill neighborhood will be the victim of a crime next year? What is the probability q that a resident will not be a victim? Solution: Using the Piton Foundation report, we take p = 111/1000 = 0.111 and q = 1 – p = 0.889

Example 16(b) – Sampling distribution of p ^ cont’d Consider the random variable Can we approximate the distribution with a normal distribution? Explain. Solution: np = 50(0.111) = 5.55 nq = 50(0.889) = 44.45 Since both np and nq are greater than 5, we can approximate the distribution with a normal distribution.

Example 16(c) – Sampling distribution of p ^ cont’d What are the mean and standard deviation for the distribution? Solution: