Question

The GPAs of all 5540 students enrolled at a university have an approximate normal distribution with a mean of 3.02 and a standard deviation of .29. Let x be the mean GPA of a random sample of 48 students selected from this university. Find the mean and standard deviation of x, and comment on the shape of its sampling distribution.​

According to a Gallup poll conducted January 5–8, 2014, 67% of American adults were dissatisfied with the way income and wealth are distributed in America. Assume that this percentage is true for the current population of American adults. Let pˆ be the proportion in a random sample of 400 American adults who hold the above opinion. Find the mean and standard deviation of the sampling distribution of pˆ and describe its shape.

Answer 1

a)

Given,

= 3.02 , = 0.29, n = 48

Mean of the sampling distribution of sample mean =   = 3.02

Standard deviation of the sampling distribution of sample mean = = / sqrt(n)

= 0.29 / Sqrt(48)

= 0.04186

Since sample size n = 48 is sufficiently large( n >= 30), shape of sampling distribution is approximately

normal (bell shaped).

b)

Given = 0.67, n = 400

Mean of the sampling distribution of = = 0.67

Standard deviation of = sqrt( ( 1 - ) / n)

= sqrt( 0.67 * 0.33 / 400)

= 0.0235

Sample is large enough (n >= 30). so shape of sampling distribution is approximately normal.