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 independent random samples of size $n = 61$ pregnancies were obtained from this population, we would expect sample(s) to have a sample mean of exactly 261 days.
b. if 100 independent random samples of size $n = 61$ pregnancies were obtained from this population, we would expect sample(s) to have a sample mean of 261 days or more.
c. if 100 independent random samples of size $n = 61$ pregnancies were obtained from this population, we would expect 1 sample(s) to have a sample mean of 261 days or less.
(e) what might you conclude if a random sample of 61 pregnancies resulted in a mean gestation period of 261 days or less?
this result would be unusual, so the sample likely came from a population whose mean gestation period is less than 266 days.
(f) what is the probability a random sample of size 20 will have a mean gestation period within 8 days of the mean?
the probability that a random sample of size 20 will have a mean gestation period within 8 days of the mean is (round to four decimal places as needed)

Explanation:

Step1: Calculate the standard error

The standard error formula is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$. Given $\sigma = 16$ and $n = 20$, we have $\sigma_{\bar{x}}=\frac{16}{\sqrt{20}}\approx3.5777$.

Step2: Find the z - scores

The lower bound $x_1=\mu - 8=266 - 8 = 258$ and the upper bound $x_2=\mu + 8=266+8 = 274$.
The z - score formula is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$.
For $x_1$: $z_1=\frac{258 - 266}{3.5777}\approx - 2.24$.
For $x_2$: $z_2=\frac{274 - 266}{3.5777}\approx2.24$.

Step3: Calculate the probability

Using the standard normal distribution table, $P(-2.24From the standard normal table, $P(Z < 2.24)=0.9875$ and $P(Z<-2.24)=0.0125$.
So $P(-2.24

Answer:

$0.9750$