Question

Suppose the lengths of the pregnancies of a certain animal are approximately normally distributed with mean H 141 days and standard deviation a (a) What is the probability that a randomly selected pregnancy lasts less than 137 days? The probability that a randomly selected pregnancy lasts less than 137 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.) O A. If 100 pregnant individuals were selected independently from this population, we would expect pregnancies to last exactly 137 days. O B. If 100 pregnant individuals were selected independently from this population, we would expect pregnancies to last less than 137 days. 。C. If 100 pregnant individuals were selected independent from this population, we would expect pregnancies as more a 37 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 x is (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 137 days or less? The probability that the mean of a random sample of 17 pregnancies is less than 137 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. 13 days. Complete parts (a) through (1) below. L _with4°Dando (Round to the nearest integer as needed.) 0 A. lf 100 independent random samples of size n-17 pregnancies were obtained from this population, we would expect O B lf 100 independent random samples of size n = 17 pregnancies were obtained rom this population, we would expect ○ C. If 100 independent random samples of size 17 pregnancies were obtained from this population, we would expect sample(s) to have a sample mean of 137 days or more sample s to have a sample mean of 137 days or less. sample(s) to have a sample mean of exactly 137 days. (d) What is the probability that a random sample of 69 pregnancies has a mean gestation period of 137 days or less? The probability that the mean of a random sample of 69 pregnancies is less than 137 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.) O A. If 100 independent random samples of size n 69 pregnancies were obtained from this population, we would expect sample(s) to have a sample mean of exactly 137 days. OB. If 100 independent random samples of size n 69 pregnancies were obtained from this population, we would expect sample(s) to have a sample mean of 137 days or less C lf 100 dependent random samples o size n-69 pregnancies were obtained from this population, we would expect (e) What might you conclude if a random sample of 69 pregnancies resulted in a mean gestation period of 137 days or less? This result would be (f) What is the probability a random sample of size 17 will have a mean gestation period within 11 days of the mean? The probability that a random sample of size 17 will have a mean gestation period within 11 days of the mean is sample s to have a sample mean of 137 days or more. V so the sample likely came from a population whose mean gestation period is 141 days Round to four decimal places as needed.)

Answer 1

Here we have : $\mu$ = 141 , $\sigma$ = 13

a) Here we need to find,

p ( x < 137 )

$=p\left ( \frac{x-\mu }{\sigma }<\frac{137-141}{13} \right )$

= p ( z < -0.31 )

= 0.3783 ------------ ( using excel formula " =norm.s.dist(-0.31,1) " )

Expected value = np = 100 * 0.3783 = 37.83

Interpretation : If 100 pregnant individuals were selected independently from this population, we would expect 37.83 pregnancies to last less than 137 days.

b) Here n = 17 .

Sampling distribution of $\overline{x}$ is normally distributed with $\mu _{\overline{x}}$ = 141 and $\sigma _{\overline{x}}=\sigma /\sqrt{n}=13/\sqrt{17}=3.1530$

c) Here we need to find

p ( $\overline{x}$ $\leq$ 137 )

$=p\left ( \frac{\overline{x}-\mu _{\overline{x}}}{\sigma _{\overline{x}}}\leq \frac{137-141}{3.1530} \right )$

= p ( z < -1.27 )

= 0.1020 -------------( using excel formula " =norm.s.dist( -1.27,1 ) " )

Interpretation : If 100 pregnant individuals of size n=17 were selected independently from this population, we would expect 10.20 samples to have sample mean of 137 or less.

d) Here n = 69

So we have  $\mu _{\overline{x}}$ = 141 and $\sigma _{\overline{x}}=\sigma /\sqrt{n}=13/\sqrt{69}=1.5650$

We need to find,

p ( $\overline{x}$ $\leq$ 137 )

$=p\left ( \frac{\overline{x}-\mu _{\overline{x}}}{\sigma _{\overline{x}}}\leq \frac{137-141}{1.5650} \right )$

= p ( z < -2.56 )

= 0.0052 -------------( using excel formula " =norm.s.dist( -2.56,1 ) " )

Interpretation : If 100 pregnant individuals of size n=69 were selected independently from this population, we would expect 0.52 samples to have sample mean of 137 or less.

e) This probability is very less.

So,

This result would be unusual, so the sample likely came from a population whose mean gestation period is less than 141 days.

f) The probability that a random sample of size 17 will have a mean gestation period within 8 days of the mean is nothing but within 8 standard deviation of mean.

So this value will be equal to 0.