Question

IV. The College Board American College Testing Program reported a population mean SAT score of 960 (The New York Times Almanac). Assume that the population standard deviation is 100. 1.What is the probability that a random sample of 75 students will provide a sample mean SAT score within 10 of the population mean? 2. What is the probability that a random sample of 75 students will provide a sample mean SAT score within 20 of the population mean?

Answer 1

1. 61.56% probability that a random sample of 75 students will provide a sample mean SAT score within 10 of the population mean

2. 91.64% probability that a random sample of 75 students will provide a sample mean SAT score within 20 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

When the distribution is normal, we use 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.

Central Limit Theorem

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

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this question, we have that:

`\mu = 960, \sigma = 100, n = 75, s = (100)/(√(75)) = 11.547`

1.What is the probability that a random sample of 75 students will provide a sample mean SAT score within 10 of the population mean?

This is the pvalue of Z when X = 960 + 10 = 970 subtracted by the pvalue of Z when X = 960 - 10 = 950. So

X = 970

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

By the Central Limit Theorem

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

`Z = (970 - 960)/(11.547)`

`Z = 0.87`

`Z = 0.87` has a pvalue of 0.8078

X = 950

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

`Z = (950 - 960)/(11.547)`

`Z = -0.87`

`Z = -0.87` has a pvalue of 0.1922

0.8078 - 0.1922 = 0.6156

61.56% probability that a random sample of 75 students will provide a sample mean SAT score within 10 of the population mean.

2. What is the probability that a random sample of 75 students will provide a sample mean SAT score within 20 of the population mean?

This is the pvalue of Z when X = 960 + 20 = 980 subtracted by the pvalue of Z when X = 960 - 20 = 940. So

X = 980

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

`Z = (980 - 960)/(11.547)`

`Z = 1.73`

`Z = 1.73` has a pvalue of 0.9582

X = 940

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

`Z = (940 - 960)/(11.547)`

`Z = -1.73`

`Z = -1.73` has a pvalue of 0.0418

0.9582 - 0.0418 = 0.9164

91.64% probability that a random sample of 75 students will provide a sample mean SAT score within 20 of the population mean.