Question

The distribution of SAT II Math scores is approximately normal with mean 660 and standard deviation 90. The probability that 100 randomly selected students will have a mean SAT II Math score greater than 670 is approximately what

Answer 1

Using the normal distribution and the central limit theorem, it is found that there is a 0.1335 = 13.35% probability that 100 randomly selected students will have a mean SAT II Math score greater than 670.

Normal Probability Distribution

In a normal distribution with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

• It measures how many standard deviations the measure is from the mean.

• After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.

• By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

In this problem:

• The mean is of 660, hence $\mu = 660$.

• The standard deviation is of 90, hence $\sigma = 90$.

• A sample of 100 is taken, hence $n = 100, s = \frac{90}{\sqrt{100}} = 9$.

The probability that 100 randomly selected students will have a mean SAT II Math score greater than 670 is 1 subtracted by the p-value of Z when X = 670, hence:

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{670 - 660}{9}$

$Z = 1.11$

$Z = 1.11$ has a p-value of 0.8665.

1 - 0.8665 = 0.1335.

0.1335 = 13.35% probability that 100 randomly selected students will have a mean SAT II Math score greater than 670.

To learn more about the normal distribution and the central limit theorem, you can take a look at https://brainly.com/question/24663213