ANSWER:
a. 269.1 and 252.9
b. 262.3 and 259.6
Explanation:
Given:
mean = 261days
standard deviation = 12days
We use the normal table to calculate the value of z, like this:
a.
![\begin{gathered} P(-z<p>Using z-score formula,</p>[tex]\begin{gathered} x=\pm z\cdot\sigma+\mu \\ x_1=0.6745\cdot12+261=269.1 \\ x_2=-0.6745\cdot12+261=252.9 \end{gathered}]()
Therefore, the middle 50% are from 269.1 and 252.9
b.
n = 37
![\begin{gathered} \sigma_{\bar{x}}=\frac{\sigma}{\sqrt[]{n}} \\ \sigma_{\bar{x}}=\frac{12}{\sqrt[]{37}} \\ \: \sigma_{\bar{x}}=1.97 \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/nctej1wg7fdr1npzrfk3wunmlqkab8aymz.png)
Using z-score formula:

Therefore, the middle 50% are from 262.3 and 259.6