Question

GRADED PROBLEM SET #5

Answer each of the following questions completely. There are a total of 20 points possible in the assignment.

1. (8 pts) Based on past results found in the Information Please Almanac there is a 0.1919 probability that a baseball World Series will last four games, a 0.2121 probability that it will last five games, a 0.2223 probability that it will last six games, and a 0.3737 probability that it will last seven games.
   1. Does the given information describe a probability distribution? Explain.

2. Assuming that the given information describes a probability distribution, find the mean and standard deviation for the number of games in World Series.

2. (4 pts) Men heights are normally distributed with a mean of 70 inches and a standard deviation of 3 inches. Doorframes have to be designed so that 99.9% of all men can pass under without stooping. What is the height of the doorframe?

3. (8 pts) Tree diameters in a plot of land are normally distributed with a mean of 14 inches and a standard deviation of 3.2 inches.
   1. What is the probability that an individual tree has a diameter between 13 inches and 16.3 inches?

2. What is the probability that an individual tree has a diameter less than 12 inches?

3. What is the probability that an individual tree has a diameter of at least 15 inches?

4. Find the cutoff for the 80^th percentile of tree diameters. (Provide the probability notation)

Answer 1

1)a) Total = 0.1919 + 0.2121 + 0.2223 + 0.3737 = 1

Since the sum of all individual probabilities is equal to 1, so it describes a probability distribution.

b) mean(E(X)) = 4 * 0.1919 + 5 * 0.2121 + 6 * 0.2223 + 7 * 0.3737 = 5.7778

E(X^2) =  4^2 * 0.1919 + 5^2 * 0.2121 + 6^2 * 0.2223 + 7^2 * 0.3737 = 34.687

Variance = E(X^2) - (E(X))^2

= 34.687 - (5.7778)^2

= 1.304

Standard deviation = sqrt(1.304) = 1.1419

2) P(X < x) = 0.999

Or, P((X - )/ < (x - )/) = 0.999

Or, P(Z < (x - 70)/3) = 0.999

Or, (x - 70)/3 = 3.08

Or, x = 3.08 * 3 + 70

Or, x = 79.24

3)a) P(13 < X < 16.3)

= P((13 -  )/ < (X -  )/ < (16.3 -  )/)

= P((13 - 14)/3.2 < Z < (16.3 - 14)/3.2)

= P(-0.31 < Z < 0.72)

= P(Z < 0.72) - P(Z < -0.31)

= 0.7642 - 0.3783

= 0.3859

b) P(X < 12)

= P((X -  )/ < (12 -  )/)

= P(Z < (12 - 14)/3.2)

= P(Z < -0.63)

= 0.2643

c) P(X > 15)

= P((X -  )/> (15 -  )/)

= P(Z > (15 - 14)/3.2)

= P(Z > 0.31)

= 1 - P(Z < 0.31)

= 1 - 0.6217

= 0.3783

d) P(X < x) = 0.8

Or, P((X -  )/ < (x -  )/) = 0.8

Or, P(Z < (x - 14)/3.2) = 0.8

Or, (x - 14)/3.2 = 0.84

Or, x = 0.84 * 3.2 + 14

Or, x = 16.688

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