Normal Approximations to Binomial Distributions Section 5.5 Normal Approximations to Binomial Distributions

Approximating a Binomial Distribution Normal Approximation to a Binomial Distribution: If np ≥ 5 and nq ≥ 5, then the binomial random variable x is approximately normally distributed, with mean 𝜇=𝑛𝑝 And standard deviation 𝜎= 𝑛𝑝𝑞

Example 1 Two binomial experiments are listed. Decide whether you can use the normal distribution to approximate x, the number of people who reply yes. If you can, find the mean and standard deviation. If you cannot, explain why. 34% of people in the US say that they are likely to make a New Year’s resolution. You randomly select 15 people in the US and ask each if he or she is likely to make a New Year’s resolution.  

Example 1 Continued Two binomial experiments are listed. Decide whether you can use the normal distribution to approximate x, the number of people who reply yes. If you can, find the mean and standard deviation. If you cannot, explain why. 6% of people in the US who made a New Year’s resolution resolved to exercise more. You randomly select 65 people in the US who made a resolution and ask each if he or she resolved to exercise more.  

Example 2 Use a correction for continuity to convert each of the following binomial intervals to a normal distribution interval. The probability of getting between 270 and 310 successes, inclusive. The probability of at least 158 successes. The probability of getting less than 63 successes. The probability of getting between 57 and 83 success, inclusive. The probability of getting at most 54 successes.

Approximating Binomial Probabilities Guidelines: Using the normal distribution to approximate binomial probabilities: Verify that the binomial distribution applies. Determine if you can use the normal distribution to approximate x, the binomial variable. Find the mean and standard deviation for the distribution. Apply the appropriate continuity correction. Shade the corresponding area under the curve. Find the corresponding z-score(s). Find the probability.

Example 3 Thirty-four percent of people in the U.S. say that they are likely to make a New Year’s resolution. You randomly select 15 people in the U.S. and ask each if he or she is likely to make a New Year’s resolution. What is the probability that fewer than eight of them respond yes? There is a 90.49% chance that fewer than 8 people will respond yes.

Example 4 Thirty-eight percent of people in the U.S. admit that they snoop in other people medicine cabinets. You randomly select 200 people in the U.S. and ask each if he or she snoops in other people’s medicine cabinets. What is the probability that at least 70 will say yes? There is an 82.89% chance that at least 70 people will admit to snooping in other people’s medicine cabinets.

Example 4 What is the probability that at most 85 people will say yes? There is an 91.62% chance that at least 85 people will admit to snooping in other people’s medicine cabinets.

TOTD A binomial experiment is given. Decide whether you can use the normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why. A survey of U.S. adults found that 44% read every word of a credit card contract. You ask 10 adults selected at random if he or she reads every word of a credit card contract.

Example 5 A survey reports that 95% of Internet users use Microsoft Internet Explorer as their browser. You randomly select 200 Internet users and ask each whether he or she uses Microsoft Internet Explorer as his or her browser. What is the probability that exactly 194 will say yes? There is a 5.5% chance that exactly 194 people will use Microsoft Internet Explorer as their browser.

Example 5 What is the probability that exactly 191 people will say yes? There is a 12.43% chance that exactly 191 people will use Microsoft Internet Explorer as their browser.

TOTD Fifty-two percent of adults say chocolate chip is their favorite cookie. You randomly select 40 adults and ask each if chocolate chip is his or her favorite cookie. Find the probability that at most 15 people say chocolate chip is their favorite cookie. Find the probability that at least 15 people say chocolate chip is their favorite cookie.