Question

A swimmer swims the 50m freestyle with a mean of 27.2 seconds and a standard deviation of 1.3 seconds. Assume the distribution is normal.

1. What proportion of the time will the swimmer’s time be between 26 and 28 seconds?

2. What proportion of the time will the swimmer’s time be less than 25 seconds?

3. What proportion of the time will the swimmer’s time be greater than 29 seconds?

4. What time represents the best 10% of all of the swimmer’s times.

5. Now let’s look at the average time for a randomly selected 30 of his swims. What kind of distribution does the sample mean have (this is a written question – give the mean and standard deviation)?

6. What is the probability that his average time for these swims is greater than 28 seconds?

7. A contractor decided to build homes that will include the middle 80% of the market. If the average size (in square feet) of homes built is 1,810, find the minimum and maximum sizes of the homes the contractor should build. Assume the standard deviation is 92 square feet and the variable is normally distributed.

Answer 1

Answer)

As the data is normally distributed we can use standard normal z table to estimate the answers

Z = (x-mean)/s.d

Given mean = 27.2

S.d = 1.3

1)

P(26<x<28) = P(x<28) - P(x<26)

P(x<28)

Z = (28 - 27.2)/1.3 = 1.04

From z table, P(z<1.04) = 0.8508

P(x<26)

Z = -1.56

From z table, P(z<-1.56) = 0.0594

Required probability is 0.8508 - 0.0594 = 0.7914

2)

P(x<25)

Z = (25-27.2)/1.3 = -1.69

From z table, P(z<-1.69) = 0.0455

3)

P(x>29)

Z = (29-27.2)/1.3 = 1.38

From z table, P(z>1.38) = 0.0838

4)

From z table, P(z<-1.28) = 10%

-1.28 = (x - 27.2)/1.3

X = 25.536