Question

Suppose the lengths of the pregnancies of a certain animal are approximately normally distributed with mean= 169days and standard deviation=14 days. complete parts (a) through (f) below.

(a) What is the probability that a randomly selected pregnancy lasts less than 164 days?

The probability that a randomly selected pregnancy lasts less than 164 days is approximately _. (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 _ pregnancies to last less than 164 days.

B. If 100 pregnant individuals were selected independently from this population, we would expect _ pregnancies to last more than 164 days.

C. If 100 pregnant individuals were selected independently from this population, we would expect _ pregnancies to last exactly 164 days.

(b) Suppose a random sample of 17 pregnancies is obtained. Describe the sampling distribution of the sample mean length of pregnancies.

The sampling distribution of sample mean is _ (skewed right, normal, or skewed left) with μ_x = _. and σ_x= _. (Round to four decimal places as needed.)

(c) What is the probability that a random sample of 17 pregnancies has a mean gestation period of 164 days or less?

The probability that the mean of a random sample of 17 pregnancies is less than 164 days is approximately _. (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 independent random samples of size n= 17 pregnancies were obtained from this population, we would expect _ sample(s) to have mean of exactly 164 days.

B. if 100 independent random samples of size n= 17 pregnancies were obtained from this population, we would expect _ sample(s) to have a sample mean of 164 days or less.

C. If 100 independent random samples of size n= 17 pregnancies were obtained from this population, we would expect _ sample(s) to have a sample mean of 164 days or more.

(d) What is the probability that a random sample of 43 pregnancies has a mean gestation period of 164 days or less?

The probability that the mean of a random sample of 43 pregnancies is less than 164 days is approximately _. (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 independent random samples of size n= 43 pregnancies were obtained from this population, we would expect _ sample(s) to have a sample mean of exactly 164 days.

B. If 100 independent random sample of size n= 43 pregnancies were obtained from this population, we would expect _ sample(s) to have a sample mean of 164 days or more.

C. If 100 independent random samples of size n= 43 pregnancies were obtained from this population, we would expect _ sample(s) to have a sample mean of 164 days or less.

(e) what might you conclude if a random sample of 43 pregnancies resulted in a mean gestation period of 164 days or less?

This result would be _ (Expected, or Unusual) so the sample likely came from a population whose mean gestation period is (Equal to, Less than, or Greater than) 169 days.

(f) What is the probability a random sample of size 18 will have a mean gestation period within 12 days of the mean?

The probability that a random sample of size 18 will have a mean gestation period within 12 days of the mean is _. (Round to four decimal places as needed.)

Answer 1

a)

for normal distribution z score =(X-μ)/σx
here mean=       μ=	169
std deviation   =σ=	14.0000

probability that a randomly selected pregnancy lasts less than 164 days :

probability =

P(X<164)

=

P(Z<-0.36)=

0.3594

A. If 100 pregnant individuals were selected independently from this population, we would expect 36 pregnancies to last less than 164 days.

b)

The sampling distribution of sample mean is normal with μ_x = 169 and σ_x= 14/sqrt(17)=3.3955

c)

probability that a random sample of 17 pregnancies has a mean gestation period of 164 days or less :

probability =

P(X<164)

=

P(Z<-1.47)=

0.0708

B. if 100 independent random samples of size n= 17 pregnancies were obtained from this population, we would expect 7 sample(s) to have a sample mean of 164 days or less.

d) probability that a random sample of 43 pregnancies has a mean gestation period of 164 days or less

probability =

P(X<164)

=

P(Z<-2.34)=

0.0096

C. If 100 independent random samples of size n= 43 pregnancies were obtained from this population, we would expect 1 sample(s) to have a sample mean of 164 days or less.

e)

This result would be Unusual so the sample likely came from a population whose mean gestation period is Less than 169 days.

f)

probability =

P(157<X<181)

=

P(-3.64<Z<3.64)=

0.9999-0.0001=

0.9998