The Central Limit Theorem

Transcription

1 Section 6-5 The Central Limit Theorem I. Sampling Distribution of Sample Mean ( ) Eample 1: Population Distribution Table P() 1/4 1/4 1/4 1/4 μ (a) Find the population mean and population standard deviation of the population distribution table. σ (b) Construct a probability histogram for

2 Eample 2: From the population distribution of eample 1, 2 random variables are randomly selected. (a) List out all possible combinations (sample space) and for each combination. (b) Construct a probability distribution table for. (c) Construct a probability histogram for.

3 (d) Find the mean of the sampling distribution of. (e) Find the standard deviation of the sampling distribution of. (f) Compare μ with μ. (g) Compare σ with σ.

4 Population parameter Sample statistics Mean μ μ Standard deviation σ σ Population Distribution Sampling Distribution P() 1 4 P() 4/10 3/10 2/10 1/ II. Central Limit Theorem If the population distribution is normally distributed, the sampling distribution of will be normally distributed for n 30. P( ) P( ) If the population distribution is not normally distributed, the sampling distribution of will be normally distributed for any size of n 30 P( ) P( ) μ = μ σ = σ n

5 Eample 1: Population distribution P( ) Given: μ = 50, σ = 10 (a) Find μ and σ for n = 4 (b) Is the sampling distribution normally distributed? (c) If n is changed from 4 to 36, is the sampling distribution normally distributed? Eample 2: (Ref: General Statistics by Chase/Bown, 4 th Ed.) A population has mean 325 and variance 144. Suppose the distribution of sample means is generated by random samples of size 36. (a) Find μ and σ (b) Find P ( < 323) (c) Find P(321< < 327)

6 Eample 3: The average number of days spent in a North Carolina hospital for a coronary bypass in 1992 was 9 days and the standard deviation was 4 days (North Carolina Medical Database Commission, Consumer s Guide to Hospitalization Charges in North Carolina Hospitals, August 1994). What is the probability that a random sample of 30 patients will have an average stay longer than 9.5 days? Eample 4: Suppose the test scores for an eam are normally distributed with μ = 75, σ = 8 (a) What percent of the students has a score greater than 85? (b) What is the probability that 4 randomly selected students will have a mean score higher than 85?

7 Section 6 6 Normal Approimation to the Binomial Distribution I. When to use a N dist. to approimate a Bi dist.? Recall that a binomial distribution is determined by n and p. When p is approimately 0.5, and as n increases, the shape of the binomial distribution becomes similar to the normal distribution. (Ref: Elementary Statistics 3 rd ed. by Bluman). In order to use a normal distribution to approimate a binomial distribution, n must be sufficiently large. It is known n will be sufficiently large if np 5 and nq 5. When using a normal distribution to approimate a binomial distribution, the mean and standard deviation of the normal distribution is the same as the binomial distribution. Now recall the formulas for finding the mean and standard deviation of a binomial distribution μ = np, σ = npq. II. Continuity Correction In addition to the condition of np 5 and nq 5, a correction for continuity is used in employing a continuous distribution ( N dist.) to approimate a discrete distribution ( Bi dist.). Warning: The continuity correction should be used only when approimating A binomial probability with a normal probability. Don t use the continuity correction with other normal probability problems. Continuity correction ± 0.5 Eample 1: Use the continuity correction to rewrite each epression: (a) Bi Dist. N Dist. (d) Bi Dist. N Dist. P( > 6) P( 1 < < 7) (b) Bi Dist. N Dist. (e) Bi Dist. N Dist. P( 3) P ( 5 10) (c) Bi Dist. N Dist. (f) Bi Dist. N Dist. P( 9) P (4 < 6)

8 III. Using a Normal Distribution to approimate a Binomial Distribution Step 1: Check whether the normal distribution can be used. ( np 5 and nq 5 ) Step 2: Find the mean μ and standard deviation σ. μ = np, σ = npq Step 3: Write the problem in probability notation, using. Step 4: Step 5: Step 6: Rewrite the problem by using the continuity correction factor. Continuity correction ± 0.5 Find the corresponding z value(s). μ z = σ Use the z table to find the center area and adjust the center area to answer the question. Eample 1: (Ref: General Statistics by Chase/Bown, 4 th Ed.) Assume that the eperiment is a binomial eperiment. Find the probability of 10 or more successes, where n = 13 and p =.4. (a) Use the Bi table (b) Using the normal approimation to the binomial.

9 Eample 2: A dealer states that 90% of all automobiles sold have air conditioning. If the dealer sells 250 cars, find the probability that fewer than 5 of them will not have air conditioning. Eample 3: In a corporation, 30% of the people elect to enroll in the financial investment program offered by the company. Find the probability that of 800 randomly selected people, between 260 and 300 inclusive have enrolled in the program.