Question

the length of human pregnancies is approximately normal with mean μ = 266 days and standard deviation σ = 16 days. complete parts (a) through (f)
(a) what is the probability that a randomly selected pregnancy lasts less than 261 days?
the probability that a randomly selected pregnancy lasts less than 261 days is approximately 0.3773
(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 38 pregnancies to last less than 261 days.
b. if 100 pregnant individuals were selected independently from this population, we would expect pregnancies to last exactly 261 days.
c. if 100 pregnant individuals were selected independently from this population, we would expect pregnancies to last more than 261 days.
(b) suppose a random sample of 38 human pregnancies is obtained. describe the sampling distribution of the sample mean length of pregnancies.
the sampling distribution of \\( \overline { x } \\) is with \\( \mu _ { \overline { x } } = \\) and \\( \sigma _ { \overline { x } } = \\)
(type integers or decimals rounded to four decimal places as needed.)

Explanation:

Step1: <Determine the mean of the sampling distribution>

According to the Central Limit Theorem, for a sample of size \(n\) from a population with mean \(\mu\), the mean of the sampling distribution of the sample mean \(\bar{x}\) is \(\mu_{\bar{x}}=\mu\). Given \(\mu = 266\) days, so \(\mu_{\bar{x}}=266\).

Step2: <Calculate the standard deviation of the sampling distribution>

The formula for the standard deviation of the sampling distribution (also known as the standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\). Here, \(\sigma = 16\) days and \(n = 38\). Then \(\sigma_{\bar{x}}=\frac{16}{\sqrt{38}}\approx\frac{16}{6.1644}\approx2.5955\).

Answer:

The sampling distribution of \(\bar{x}\) is approximately normal with \(\mu_{\bar{x}} = 266\) and \(\sigma_{\bar{x}}\approx2.5955\)