6.6 NORMAL APPROXIMATION TO ^ TO PBINOMIAL DISTRIBUTION ANDDISTRIBUTION Chapter 6: Normal Curves and Sampling Distributions
Page 308 – 309Normal Approximation to the BinomialDistribution Under the conditions stated below, the normal distribution can be used to approximate the binomial distribution. Consider a binomial distribution where n = number of trials r = number of successes p = probability of success on a single trial q = 1 – p = probability of failure on a single trial If np > 5 and nq > 5, then r has a binomial distribution that is approximated by a normal distribution with and Note: as n increases, the approximation becomes better
PageExample 14 – Binomial Distribution 309Graphs Notice that as n increases, the normal approximation to the binomial distribution improves
Page 311Continuity Correction The normal distribution is for a continuous random variable. The binomial distribution is for a discrete random variable. So, in order to use the normal distribution to approximate the binomial distribution, we need to make a continuity correction.
How to Make the ContinuityCorrection Convert the discrete random variable r (number of successes) to the continuous normal random variable x by doing the following: 1. If r is a left point of an interval, subtract 0.5 to obtain the corresponding normal variable x. x = r – 0.5 2. If r is a right point of an interval, ass 0.5 to obtain the corresponding normal variable x. x = r + 0.5
How to Make the Continuity Correction Example:P(6 ≤ r ≤ 10) would be approximated by P(5.5 ≤ r ≤ 10.5)
Not in Textbook!How to Find ProbabilitiesGiven a binomial distribution where n = number of trials r = number of successes p = probability of success on a single trial q = 1 – p = probability of failure on a single trial np > 5 nq > 51. Define what you are trying to find2. Make the continuity correction3. Convert to z scores Note in order to do this, you must find μ and σ.4. Use the standard normal distribution to find the corresponding probabilities
PageExample 15 – Normal 310Approximation The owner of a new apartment building must install 25 water heaters. From past experience in other apartment buildings, she knows that Quick Hot is a good brand. A Quick Hot heater is guaranteed for 5 years only, but from the owner’s past experience, she knows that the probability it will last 10 years is 0.25.
Example 15 – Normal Approximation a) What is the probability that 8 or more of the 25 water heaters will last at least 10 years? Define success to mean a water heater that lasts at least 10 years.Solution: We want: = P(r≥7.5)n = 25 P(r≥8)r = binomial randomvariablecorresponding to the =number Normalcdf(.58, E99) = .280957of successes ≈ .2810p = 0.25q = 0.75 The probability that 8 or more of the 25 waternp = 6.25 heaters will last at least 10 years is approximately
PageSampling Distributions for the ^ 313Proportion p Given n = number of binomial trials (fixed constant) r = number of successes p = probability of success on each trial q = 1 – p = probability of failure on each trial If np > 5 and nq > 5, then the random variable can be approximated by a normal random variable (x) with mean and standard deviation
Sampling Distributions for the ^Proportion p The standard error for the distribution is the standard deviation We do not use a continuity correction for the distribution. is an unbiased estimator for p, the population proportion of success.
PageExample 16 – Sampling Distribution ^ 313of 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). 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 – Sampling Distribution^of p a) 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:
Example 16 – Sampling Distribution^of p b) Consider the random variable Can we approximate the distribution with a normal distribution? Explain.Solution: Since both np and nq are greater than 5, we can approximate the distribution with a normal distribution.
Example 16 – Sampling Distribution^of p c) What are the mean and standard deviation for the distribution?Solution: