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MATH 221

MATH 221. Integrated Learning System Week 4 Lecture The Central Limit Theorem; Normal Approximations to the Binomial Distribution. Definition.

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MATH 221

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  1. MATH 221 Integrated Learning System Week 4 Lecture The Central Limit Theorem; Normal Approximations to the Binomial Distribution

  2. Definition A sampling distribution is a probability distribution of sample statistics that is formed when samples of size n are repeatedly taken from a population. If the sample statistic is the mean then the distribution is called the sampling distribution of the sample means.

  3. Properties of Sampling Distributions • The mean of the sampling distribution of the sample means x is equal to the population mean . • The standard deviation of the sampling distribution of the sample means x is equal to the population standard deviation  divided by the square foot of the sample size n. x is called the standard error of the mean.

  4. Example Four people in a carpool paid the following amounts for textbooks this semester: $120, $140, $180, and $200. Find the mean and standard deviation of the above population. List all possible samples of size two from the population. Find the mean and standard deviation of the sampling distribution of the means and compare them to the population parameters.

  5. Solution

  6. Using Technology: TI 83 How did we get the values in the previous slides using the TI 83? You may not be familiar with the lists feature of the TI 83 that allows you to input data for analysis. Follow the instructions in the notes below to see how this feature was used in the previous slides.

  7. The Central Limit Theorem If a random sample is drawn from any population, the sampling distribution of the sample means is approximately normal for a sufficiently large sample size. The larger the sample size, the more closely the sampling distribution of the sample means will resemble a normal distribution. In general, unless the underlying population has a normal distribution, a sample size of n > 30 is recommended.

  8. Probabilities and the Central Limit Theorem

  9. Example The population mean annual salary for plumbers is  = $32,500. A random sample of 42 plumbers is drawn from this population. What is the probability that the mean salary of this sample is less than $30,000? (Assume that  = $5,600)

  10. Solution Looking up z = -2.89 in a standard normal table we see that P(x < $30,000) = 0.0019.

  11. Normal Approximation to Binomial Distributions If np  5 and nq  5, then the binomial random variable x is approximately normally distributed with mean  = np and

  12. Correction for Continuity P(x = c)  P(c – 0.5 < x < c + 0.5)

  13. Process • Verify that the binomial distribution applies. Specify n, p, and q. • Determine if the normal approximation can be used. np > 5, and nq > 5. • Find the mean and the standard deviation using the formulas. • Apply the continuity correction. • Find the z-scores for c – 0.5, and c + 0.5. • Find the probability.

  14. Example • Thirty-four percent of Americans have type A+ blood. You randomly select 32 Americans and ask them if their blood type is A+. • Find the probability that 12 people say that they have A+ blood. • Find the probability that at least 12 people say they have A+ blood. • Find the probability that fewerthan 12 people say they have A+ blood.

  15. Example (Continued) • A blood drive would like to get at least 60 people with type A+ blood. If there are 150 donors, what is the probability that there will be enough type A+ blood donors?

  16. Solution So the normal approximation to the binomial distribution can be used.

  17. Solution (Part 1) c = 12 z1 = (11.5 – 10.88)/2.68 = 0.231 z2 = (12.5 – 10.88)/2.88 = 0.604 P(11.5 < x < 12.5) = 0.136 [used TI 83]

  18. Solution (Part 2) P(x 12) = 1- P(x < 11.5) = 1 – 0.591 = 0.409. [Used TI 83 to calculate P(x < 11.5)]

  19. Solution (Part 3) This part is easy since we have already calculated the value we need in Part 2. P(x < 12) = 0.59.

  20. Solution (Part 4) We want to know the probability of having at least 60 A+ donors. That probability is 1 – P(x < 59.5). 1 – P(x < 59.6) = 1 – 0.929 = 0.071.

  21. Exercise • Fifty-two percent of adults say chocolate chip are their favorite cookie. You randomly select 40 adults and ask each if chocolate chip is their 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. • Find the probability that more than 15 people say chocolate chip is their favorite cookie. • (Continued on next slide)

  22. Exercise (Continued) • A community bake sale has prepared 350 chocolate chip cookies. If the bake sale attracts 650 customers and the each buy one cookie, what is the probability that there will be enough chocolate chip cookies?

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