Question

Suppose the lengths of the pregnancies of a certain animal are approximately normally distributed with mean mu equals 194 days and standard deviation sigma equals 13 days. Complete parts​ (a) through​ (f) below. ​(a) What is the probability that a randomly selected pregnancy lasts less than 190 ​days? The probability that a randomly selected pregnancy lasts less than 190 days is approximately . 3783. ​(Round to four decimal places as​ needed.) Interpret this probability. Select the correct choice below and fill in the answer box within your choice. ​(Round to the nearest integer as​ needed.) A. If 100 pregnant individuals were selected independently from this​ population, we would expect nothing pregnancies to last more than 190 days. B. If 100 pregnant individuals were selected independently from this​ population, we would expect 38 pregnancies to last less than 190 days. Your answer is correct.C. If 100 pregnant individuals were selected independently from this​ population, we would expect nothing pregnancies to last exactly 190 days. ​(b) Suppose a random sample of 24 pregnancies is obtained. Describe the sampling distribution of the sample mean length of pregnancies. The sampling distribution of x overbar is ▼ with mu Subscript x overbarequals nothing and sigma Subscript x overbarequals nothing. ​(Round to four decimal places as​ needed.)

Answer 1

Solution:

    We are given that: the lengths of the pregnancies of a certain animal are approximately normally distributed with mean mu equals 194 days and standard deviation sigma equals 13 days.

Thus       and  

Part a) What is the probability that a randomly selected pregnancy lasts less than 190 ​days?

That is find: P( X < 190) = ........?

Find z score:

Thus we get:

P( X < 190) = P( Z < -0.31)

Look in z table for z = -0.3 and 0.01 and find area.

Thus from z table we get:

P( Z < -0.31) = 0.3783

Thus

P( X < 190) = P( Z < -0.31)

P( X < 190) = 0.3783

Thus the probability that a randomly selected pregnancy lasts less than 190 days is approximately 0.3783

Interpret this probability

B. If 100 pregnant individuals were selected independently from this​ population, we would expect 38 pregnancies to last less than 190 days.

Part b) Suppose a random sample of 24 pregnancies is obtained. Describe the sampling distribution of the sample mean length of pregnancies.

Since we are given that: the lengths of the pregnancies of a certain animal are approximately normally distributed with mean mu equals 194 days and standard deviation sigma equals 13 days

then sampling distribution of the sample mean length of pregnancies is also approximately normally distributed

with

and

that is: