Question

37% of Americans say that math is the most important subject in school. In a random sample of 400 Americans, what is the probability that between 40% and 45% will say that math is the most important subject?

Answer 1

0.1070 = 10.70% probability that between 40% and 45% will say that math is the most important subject

Explanation:

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

Normal Probability Distribution:

Problems of normal distributions 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.

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.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

37% of Americans say that math is the most important subject in school.

This means that
`p = 0.37`

Sample of 400 Americans

This means that
`n = 400`

Mean and standard deviation:

`\mu = p = 0.37`

`s = \sqrt{(p(1-p))/(n)} = \sqrt{(0.37*0.63)/(400)} = 0.0241`

What is the probability that between 40% and 45% will say that math is the most important subject?

This is the pvalue of Z when X = 0.45 subtracted by the pvalue of Z when X = 0.4. So

X = 0.45

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

By the Central Limit Theorem

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

`Z = (0.45 - 0.37)/(0.0241)`

`Z = 3.31`

`Z = 3.31` has a pvalue of 0.9995

X = 0.4

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

`Z = (0.4 - 0.37)/(0.0241)`

`Z = 1.24`

`Z = 1.24` has a pvalue of 0.8925

0.9995 - 0.8925 = 0.1070

0.1070 = 10.70% probability that between 40% and 45% will say that math is the most important subject