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)
(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 0.0268
(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.)
oa. if 100 independent random samples of size n = 38 pregnancies were obtained from this population, we would expect sample(s) to have a sample mean of 261 days or more.
ob. if 100 independent random samples of size n = 38 pregnancies were obtained from this population, we would expect sample(s) to have a sample mean of exactly 261 days.
c. if 100 independent random samples of size n = 38 pregnancies were obtained from this population, we would expect 3 sample(s) to have a sample mean of 261 days or less
(d) what is the probability that a random sample of 61 pregnancies has a mean gestation period of 261 days or less?
the probability that the mean of a random sample of 61 pregnancies is less than 261 days is approximately
(round to four decimal places as needed.)

Explanation:

Step1: Calculate the standard error

The formula for the standard error of the mean is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$. Given $\sigma = 16$ and $n = 61$, we have $\sigma_{\bar{x}}=\frac{16}{\sqrt{61}}\approx2.05$.

Step2: Calculate the z - score

The z - score formula is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$. Here, $\bar{x} = 261$, $\mu=266$, so $z=\frac{261 - 266}{2.05}=\frac{-5}{2.05}\approx - 2.44$.

Step3: Find the probability

Using the standard normal distribution table (or a calculator with a normal - distribution function), we find $P(Z\leq - 2.44)$. Looking up the value in the standard normal table, $P(Z\leq - 2.44)=0.0073$.

Answer:

$0.0073$