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M. Continuous Distributions.

1. Introduction.

These are optional! 6.28-6.32 [6.29 – 6.33] (6.24 - 6.28)

2. Properties of the Normal Distribution.

Text 6.1-6.2, 6.4!, 6.5a-c [6.1, 6.2, 6.5a-e.] (6.1, 6.2, 6.5a-e.) M1 a-g, M2, M3.

3. Percentiles and Intervals about the Mean.

Text 6.5d, 6.6!, 6.8 [6.4, 6.5f-h, 6.8.] (6.4, 6.5f-h, 6.8.) M4, M1 h-j. Graded assignment 4. M8.

4. Normal Approximation to the Binomial Distribution.

6.41 [6.57*], M6, M7.

5. Normal Approximation to the Poisson Distribution.

? [6.60*], M5.

Exercise 6.60(Not in 8th edition): The number of cars arriving per minute at a toll booth is Poisson distributed with a mean of 2.5. What is the probability that in any given minute: a. No cars arrive.

b. Not more than 2 cars arrive? c. What is the approximate probability that in a ten minute period not more than 20 cars arrive? d. What is the approximate probability that in a ten minute period between 20 and 30 cars arrive?

6. Review of Conditions for Approximation of One Distribution by Another.

This document includes solutions to sections 4 and 5. is used to signal a transition to the Normal distribution.

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Normal approximation to Binomial Problems

Exercise 6.57 in 9th (Not in 8th edition): An airline has 3 different choices on its dessert menu. Each dessert is equally likely to be chosen.

a. If a random sample of 4 passengers is chosen, what is the probability that at least 2 will choose ice cream for dessert?

b. If a random sample of 21 passengers is chosen, what is the approximate probability that at least 2 will choose ice cream for dessert?

Solution: a) Use the binomial distribution, , with and . .

b) . Note that now and . These are both above 5 so we can use a Normal distribution.

=.9946.


Exercise 6.41 in 10th (Not in 8th or 9th edition): An airline has 3 different choices on its dessert menu. Each dessert is equally likely to be chosen.

a. If a random sample of 90 passengers is chosen, what is the approximate probability that at least 20 will choose ice cream for dessert?

b. If a random sample of 90 passengers is chosen, what is the approximate probability that exactly 20 will choose ice cream for dessert?

c. If a random sample of 90 passengers is chosen, what is the approximate probability that fewer than 20 will choose ice cream for dessert? (Watch your English!)

Solution: and . So and and we can use the Normal distribution. and

a)

b)

c)

Problem M6: Do the following binomial problems using the Poisson or the Normal distribution as appropriate:

a. If and , find.

b. If and , find

Solution: a) First, test to see if we can substitute the Normal distribution in place of the binomial distribution. I tested to see if and . Since , and is not large enough and we cannot use the Normal distribution.

Second, test to see if we can use the Poisson distribution in place of the binomial distribution. I tested to see if Since is above 500, we can use the Poisson distribution with a parameter of From the Poisson table

b) First, test to see if we can substitute the Normal distribution in place of the binomial distribution. I tested to see if and . Since , and is just large enough and we can use the Normal distribution with and If we use the continuity correction,

Second, Test to see if we can use the Poisson distribution in place of the binomial distribution. . I tested to see if Since is below 500, we cannot use the Poisson distribution.


Problem M7 (Naggar): We send a survey out to 200 people. The probability that any one person will return it is 0.1. What is the chance of between 23 and 33 returns? More than 100 returns?

Solution: First, test to see if we can substitute the Normal distribution in place of the binomial distribution. I tested to see if and . Since , and is large enough and we can use the Normal distribution with and If we use the continuity correction, Also

.

Second, test to see if we can use the Poisson distribution in place of the binomial distribution. I tested to see if Since is above 500, we can use the Poisson distribution with a parameter of From the Poisson table Also

. Note that since , the cumulative probability for any higher number must be one too.

Normal approximation to Poisson Problems

.

Exercise 6.60 in 9th (Not in 8th or 10th edition): The number of cars arriving per minute at a toll booth is Poisson distributed with a mean of 2.5. What is the probability that in any given minute: a. No cars arrive.

b. Not more than 2 cars arrive? c. What is the approximate probability that in a ten minute period not more than 20 cars arrive? d. What is the approximate probability that in a ten minute period between 20 and 30 cars arrive?

Solution: a) The Poisson table with parameter 2.5 says

b) =.91970. c) In a 10-minute period . We are expected to get

. Actually, our Poisson table gives .18549. d) and the Poisson table says


Problem M5: Assume that x has the Poisson distribution with a parameter of 25. Find using both the Poisson and the Normal distributions. Compare the answers.

Solution: Since the mean is 25, the variance is also 25 and the standard deviation is the square root of 25, or 5. If we do this without a continuity correction, we have

If we insist on a continuity correction and explicitly include the lower limit (Usually unnecessary), we find the Poisson table says that .

Problem M5a: Assume that has the Binomial distribution with and . Find .

Solution: Probably we would like to do this as a Poisson problem with , but we would be on shaky ground since is less than 500. However the expected number of successes is 25 and the expected number of failures is . Since both are over 5, we can use the Normal distribution with and . Without the continuity correction, With the continuity correction, .