Question

The scores of students on the SAT college entrance examinations at a certain high school had a normal distributionwith mean / = 555.6 and standard deviation o= 25.4.

Answer 1

We have that the scores of students on the SAT college entrance examination had a normal distribution with the following parameters:

$\begin{gathered} \mu=555.6 \\ \\ \sigma=25.4 \end{gathered}$

And we need to find the following information:

1. The probability that a student is randomly chosen from all those taking the test scores 559 or higher.

2. If we take a simple random sample of 30 students, we need to find:

• The mean of the sampling distribution

,

• The standard deviation of the sampling distribution

,

• The z-score that corresponds to the mean score x-hat = 559

,

• The probability that the mean score x-hat of these students is 559 or higher.

To find those answers, we can proceed as follows:

1. To find the probability that a student is randomly chosen from all those taking the test scores 559 or higher, we can find that probability by using the z-score associated with x = 559 as follows:

$\begin{gathered} z=\frac{x-\mu}{\sigma} \\ \\ z=\frac{559-555.6}{25.4}=0.133858267717\approx0.1339 \\ \end{gathered}$

And now, we can use the cumulative standard normal distribution table to find the cumulative probability for z = 0.1339. Then we have:

$P(z<0.1339)=0.553259174857$

Since we need test scores of 559 or higher, then we have:

$\begin{gathered} P(z>0.1339)=1-P(z<0.1339)=0.446740825143\approx0.4467 \\ \end{gathered}$

Therefore, the probability for test scores of 559 or higher is about 0.4467.