Please show the sampling distribution as well Scores on a na

Please show the sampling distribution as well!

Scores on a national writing exam are approximately normally distributed, with a mean of 490 and a standard deviation of 20. Consider random samples of size 25 from the population of test takers.

a) Describe the sampling distribution of the sample mean.

Shape: ___________________________

Mean =
Standard Deviation =

Sketch the sampling distribution of the sample mean.

b) Find the probability that a random sample of 25 test takers will have a mean exam score that is greater than 496?

c) Find k such that P( x < k) = .01.

Solution

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Scores on a national writing exam are approximately normally distributed, with a mean of 490 and a standard deviation of 20. Consider random samples of size 16 from the population of test takers.

b) Find the probability that a random sample of 16 test takers will have a mean exam score that is greater than 498?

c) Find k such that P(x_bar< k) = .025.

Answer

(b) P(xbar> 498) = P((xbar-mean)/(s/vn) >(498-490 )/(20/4))

=P(Z>1.6) =0.0548 (from standard normal table)

(c) P(x_bar< k) = .025

--> P(Z<(k-490 )/(20/4) ) = 0.025

--> (k-490 )/(20/4) =-1.96 (from standard normal table)

--> k = 490-1.96*(20/4)

=480.2