Question

The average playing time of compact discs in a large collection is 37 min, and the standard deviation is 5 min.

(a) What value is 1 standard deviation above the mean? 1 standard deviation below the mean? What values are 2 standard deviations away from the mean?

1 standard deviation above the mean ___

1 standard deviation below the mean ____

2 standard deviation above the mean ____

2 standard deviation below the mean _____

(b) Without assuming anything about the distribution of times, at least what percentage of the times are between 27 and 47 min? (Round the answer to the nearest whole number.)

At least  

_____

(c) Without assuming anything about the distribution of times, what can be said about the percentage of times that are either less than 22 min or greater than 52 min? (Round the answer to the nearest whole number.)

No more than  

%______

(d) Assuming that the distribution of times is normal, approximately what percentage of times are between 27 and 47 min? (Round the answers to two decimal places, if needed.)

_____

%

Less than 22 min or greater than 52 min?

___

%

Less than 22 min?

____

%

Answer 1

We have been given params of normal distribution. You may want to refer to just the empirical rule of normal distribution and the Chebeshev' Rule to answer the following:

Average = 37
Stdev = 5

a.

1 standard deviation above mean means that above 42(37+5) we have 16% of all playing times greater than 42

2 standard deviation above mean means that above 47(37+10) we have 2.5% of all playing times greater than 47

1 standard deviation above mean means that above 32(37-5) we have 16% of all playing times smaller than 32

1 standard deviation above mean means that above 27(37-10) we have 2.5% of all playing times smaller than 27

b. Thats 2 deviations from mean . Chebeshev' Rule says that atleast 1-1/k^2 ( where k = deviation) area in under curve within k deviations

= 1-1/2^2 = 75%

c. <22 or >52 is more area under curve beyond 3 deviations, which essentially is 1-area under 3 deviation
= 1-.997 = .003 or .3%

d. Assuming normal dist, we have 2 deviations worth area under curve within 27 and 47 min = 95%

Again, we have 3 deviations between 22 and 52, so beyond these 2 we have 1-.997 = .003 or .3%

Less than 22 min , we have .15% ( half of the above)