Question

1. Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 234 feet and a standard deviation of 58 feet. We randomly sample 49 fly balls.

b) What is the probability that the 49 balls traveled an average of less than 226 feet? (Round your answer to four decimal places.)

c) Find the 60th percentile of the distribution of the average of 49 fly balls. (Round your answer to two decimal places.)

4.The length of songs in a collector's iTunes album collection is uniformly distributed from two to 3.7 minutes. Suppose we randomly pick five albums from the collection. There are a total of 44 songs on the five albums.
d) Give the distribution of X (Round your answers to four decimal places.)

X ~ N ( 2.85 , ? )

e) Find the first quartile for the average song length. (Round your answer to two decimal places.)

f) Find the IQR (interquartile range) for the average song length. (Round your answer to two decimal places.)

6.The percent of fat calories that a person consumes each day is normally distributed with a mean of about 34 and a standard deviation of about ten. Suppose that 25 individuals are randomly chosen. Let

X = average percent of fat calories.

(a) Give the distribution of X. (Round your standard deviation to two decimal places.)

X ~ N ( 34, ? )

(c) Find the first quartile for the average percent of fat calories. (Round your answer to two decimal places.)

Salaries for teachers in a particular elementary school district are normally distributed with a mean of $44,000 and a standard deviation of $6,300. We randomly survey ten teachers from that district. (Round your answers to the nearest dollar.)

(a) Find the 90^th percentile for an individual teacher's salary.

(b) Find the 90^th percentile for the average teacher's salary.

Answer 1

a)

X ~ N(234, 58)

By Central limit theorem,   ~ N(234, 58/ = 8.2857)

b)

Probability that the 49 balls traveled an average of less than 226 feet = P( < 226)

= P[Z < (226 - 234)/8.2857]

= P[Z < -0.9655]

= 0.1671

c)

Z value for 60th percentile = 0.2533

60th percentile = 234 + 0.2533 * 8.2857 = 236.10 feet

4.

d.

Let Y be the length of songs in a collector's iTunes album collection. Y ~ Uniform(2, 3.7)

Mean of Y, E(Y) = (2 + 3.7)/2 = 2.85

Standard deviation of Y = (3.7 - 2) / = 0.4907477 (By standard deviation formula of Uniform distribution)

Standard error of mean = 0.4907477 / = 0.2195

The distribution of X (Mean of Y) is,

X ~ N ( 2.85 , 0.2195 )

e.

Z value for 1st quartile is -0.6745

First quartile for the average song length = 2.85 - 0.6745 * 0.2195 = 2.70 minutes

f.

Z value for 3rd quartile is 0.6745

Third quartile for the average song length = 2.85 + 0.6745 * 0.2195 = 3.00 minutes

IQR = Third quartile - First quartile = 3.00 - 2.70 = 0.30 minutes

6.

(a)

Standard error of mean = 10 / = 2

The distribution of X is,

X ~ N ( 34, 2 )

(c)

Z value for 1st quartile is -0.6745

First quartile for the average percent of fat calories = 34 - 0.6745 * 2 = 32.65

(a)

Z value for 90th percentile is 1.28

90^th percentile for an individual teacher's salary = $44,000 + 1.28 * $6,300 = $52064

(b)

Standard error of mean = 6300 / = 1992.235

90^th percentile for an average teacher's salary = $44,000 + 1.28 * $1992.235 = $46550.06