Question

a. If X is a normal random variable with mean 10, and if the probability that X is less than 11.54 is .72 then what is the standard deviation of X?

1.75

3.50

4.20

12.25

b. If the standard deviation of a population is 36 and we take a sample of size 9, then the standard error (the standard deviation of the sample mean) is

12.00

3.00

108.00

4.00

c. According to the empirical rule, in a normal distribution about what proportion of values falls within 3 standard deviations from the mean?

99.7%

at least 89%

4.7%

2.35%

d. The mean of the sampling distribution of sample means is the same as the population mean

True

False

Answer 1

a. Let the standard deviation be s.

Thus, X ~ N(10, s) i.e. (X - 10)/s ~ N(0,1).

P(X < 11.54) = P[(X - 10)/s < (11.54-10)/s] = P[(X - 10)/s < 1.54/s] = (1.54/s) [(.) is the cdf of N(0,1)].

Given that, (1.54/s) = 0.72 i.e. 1.54/s = (0.72) = 0.583 i.e. s = 1.54/0.583 = 2.64.

b. Standard error = = 36/3 = 12.

c. 99.7% of the values fall within 3 standard deviations of the mean.

d. The mean of the sampling distribution of sample means is same as the population mean.