Question

Suppose the lengths of the pregnancies of a certain animal are approximately normally distributed with mean mu equals 248 days and standard deviation sigma equals 24 days. Complete parts​ (a) through​ (f) below. ​(a) What is the probability that a randomly selected pregnancy lasts less than 239 ​days? The probability that a randomly selected pregnancy lasts less than 239 days is approximately nothing. ​(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 239 days. B. If 100 pregnant individuals were selected independently from this​ population, we would expect nothing pregnancies to last less than 239 days. C. If 100 pregnant individuals were selected independently from this​ population, we would expect nothing pregnancies to last exactly 239 days. ​(b) Suppose a random sample of 18 pregnancies is obtained. Describe the sampling distribution of the sample mean length of pregnancies. The sampling distribution of x overbar is ▼ skewed left normal skewed right with mu Subscript x overbarequals nothing and sigma Subscript x overbarequals nothing. ​(Round to four decimal places as​ needed.) ​(c) What is the probability that a random sample of 18 pregnancies has a mean gestation period of 239 days or​ less? The probability that the mean of a random sample of 18 pregnancies is less than 239 days is approximately nothing. ​(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 nequals18 pregnancies were obtained from this​ population, we would expect nothing ​sample(s) to have a sample mean of 239 days or less. B. If 100 independent random samples of size nequals18 pregnancies were obtained from this​ population, we would expect nothing ​sample(s) to have a sample mean of 239 days or more. C. If 100 independent random samples of size nequals18 pregnancies were obtained from this​ population, we would expect nothing ​sample(s) to have a sample mean of exactly 239 days. ​(d) What is the probability that a random sample of 49 pregnancies has a mean gestation period of 239 days or​ less? The probability that the mean of a random sample of 49 pregnancies is less than 239 days is approximately nothing. ​(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 nequals49 pregnancies were obtained from this​ population, we would expect nothing ​sample(s) to have a sample mean of exactly 239 days. B. If 100 independent random samples of size nequals49 pregnancies were obtained from this​ population, we would expect nothing ​sample(s) to have a sample mean of 239 days or more. C. If 100 independent random samples of size nequals49 pregnancies were obtained from this​ population, we would expect nothing ​sample(s) to have a sample mean of 239 days or less. ​(e) What might you conclude if a random sample of 49 pregnancies resulted in a mean gestation period of 239 days or​ less? This result would be ▼ expected, unusual, so the sample likely came from a population whose mean gestation period is ▼ equal to greater than less than 248 days. ​(f) What is the probability a random sample of size 15 will have a mean gestation period within 12 days of the​ mean? The probability that a random sample of size 15 will have a mean gestation period within 12 days of the mean is nothing. ​(Round to four decimal places as​ needed.)

Answer 1

a)

for normal distribution z score =(X-μ)/σx
here mean=       μ=	248
std deviation   =σ=	24.0000

probability that a randomly selected pregnancy lasts less than 239 ​days :

probability =

P(X<239)

=

P(Z<-0.38)=

0.3520

B. If 100 pregnant individuals were selected independently from this​ population, we would expect 35 pregnancies to last less than 239 days

b)

The sampling distribution of x overbar is normal with mean =248 and std error of mean =5.6569

c)

probability =

P(X<239)

=

P(Z<-1.59)=

0.0559

A. If 100 independent random samples of size nequals18 pregnancies were obtained from this​ population, we would expect 6 ​sample(s) to have a sample mean of 239 days or less.

d)

probability =

P(X<239)

=

P(Z<-2.63)=

0.0043

C. If 100 independent random samples of size nequals49 pregnancies were obtained from this​ population, we would expect 0 sample(s) to have a sample mean of 239 days or less

e) This result would be unusual, so the sample likely came from a population whose mean gestation period is less than 248 days.

f)

probability =

P(236<X<260)

=

P(-1.94<Z<1.94)=

0.9738-0.0262=

0.9476