Question

the length of human pregnancies is approximately normal with mean \\( \mu = 266 \\) days and standard deviation \\( \sigma = 16 \\) days. complete parts (a) through (f)
(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 normal with \\( \mu _ { \overline { x } } = 266 \\) and \\( \sigma _ { \overline { x } } = 2.5955 \\)
(type integers or decimals rounded to four decimal places as needed.)
(c) what is the probability that a random sample of 38 pregnancies has a mean gestation period of 261 days
or less?
the probability that the mean of a random sample of 38 pregnancies is less than 261 days is approximately \\( \square \\)
(round to four decimal places as needed.)

Explanation:

Step1: Calculate the z - score

The formula for the z - score of a sample mean is \(z=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}}\). We know that \(\bar{x} = 261\), \(\mu_{\bar{x}}=266\), and \(\sigma_{\bar{x}} = 2.5955\).

$$z=\frac{261 - 266}{2.5955}=\frac{- 5}{2.5955}\approx - 1.93$$

Step2: Find the probability using the standard normal distribution

We want to find \(P(\bar{X}<261)\), which is equivalent to \(P(Z < - 1.93)\) using the standard normal distribution table (or a calculator with a normal - distribution function).
Looking up \(z=-1.93\) in the standard normal table, we find that \(P(Z < - 1.93)=0.0268\)

Answer:

\(0.0268\)