Assume the sample variances to be continuous measurements. Find the probability that a random sample of 36 observations, from a normal population with variance sigma^2 = 6, will have a sample variance S^2 a) greater than 9.1;
b) between 3.5 and 10.5.

0.02566

0.972559

Step-by-step explanation:

Given that :

Sample size (n) = 36

σ² = 6

sample variance S^2

a) greater than 9.1;

b) between 3.5 and 10.5.

The degree do freedom (df) = n - 1

S² > 9.1

P(S² > 9.1) = X² > ((n - 1) * S²) / 6

P(S² > 9.1) = ((36 - 1) * 9.1) / 6

P(S² > 9.1) = (35 * 9.1) / 6

P(S² > 9.1) = 53.083

P(X² > 53.08) = 0.02566 ( chi square distribution calculator)

b) between 3.5 and 10.5.

((36 - 1) * 3.5) / 6 ≤ S² ≤ ((36 - 1) * 10.5) / 6

20.416666 ≤ S² ≤ 61.25

Using the Chisquare distribution calculator :

P(X² > 20.42) = 0.9765

P(X² > 61.25) = 0.003941

Hence,

0.9765 - 0.003941 = 0.972559

Jon can do more push ups faster based on information provided.  

Step-by-step explanation:

In order to compare both Jon and Paul find a common rate at which the can do push ups.  In our problem let's use minutes as the rate.

Jon (rate in minutes):

64/4 or 16 push ups/minute

Paul (rate in minutes)

45/3 or 15 push ups/minute

Thus based on rate per minute Jon can do one (1) more than Paul per minute.