Question

The length of time taken on the SAT for a group of students is normally distributed with a mean of 2.5 hours and a standard deviation of 0.25 hours. A sample size of n = 60 is drawn randomly from the population. Find the probability that the sample mean is between two hours and three hours.

Short answer

The probability that the sample mean is between two hours and three hours is$P(2<\overline{X}<3)=1.$

Step-by-step solution

Given information

Given:

Mean $=2.5\mathrm{h}$

Standard deviation$=0.25\mathrm{h}$

$n=60$

Formula used:

$\overline{X}~N\left({\mu }_x,\frac{ax}{\sqrt{n}}\right)$

Find the sample mean is likely to be between two and three hours.

The probability that the sample mean is between two hours and three hours is

$$\begin{gathered} \overline{X}~N\left(2.5,\frac{0.25}{\sqrt{60}}\right) \\ P(2<\overline{X}<3)=\text{normalcdf}\left(2,3,2.5,\frac{0.25}{\sqrt{60}}\right) \\ P(2<X<3)=1 \end{gathered}$$