Question

7. At a city high school, past records indicate that the MSAT scores of students have a mean of 510 and a standard deviation of 90. One hundred students in the high school are to take the test. What is the probability that the sample mean score will be (a) within 10 of the population mean

Answer 1

73.30% probability that the sample mean score will be within 10 of the population mean

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that
`\mu = E(X)`,
`\sigma = √(V(X))`.

Central limit theorem:

The Central Limit Theorem estabilishes that, for a random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sample means with size n of at least 30 can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`

In this problem, we have that:

`\mu = 510, \sigma = 90, n = 100, s = (90)/(√(100)) = 9`

What is the probability that the sample mean score will be (a) within 10 of the population mean

This is the pvalue of Z when X = 510 + 10 = 520 subtracted by the pvalue of Z when X = 510 - 10 = 500.

X = 520

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (520 - 510)/(9)`

`Z = 1.11`

`Z = 1.11` has a pvalue of 0.8665

X = 500

`Z = (X - \mu)/(s)`

`Z = (500 - 510)/(9)`

`Z = -1.11`

`Z = -1.11` has a pvalue of 0.1335

0.8665 - 0.1335 = 0.7330

73.30% probability that the sample mean score will be within 10 of the population mean