Question

The mayor of a town has proposed a plan for the construction of a new community. A political study took a sample of 1300 voters in the town and found that 45% of the residents favored construction. Using the data, a political strategist wants to test the claim that the percentage of residents who favor construction is more than 42%. Find the value of the test statistic. Round your answer to two decimal places.

Answer 1

The value of the test statistic is t = 2.19.

Explanation:

Central Limit Theorem

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)}`

Our test statistic is:

`t = (X - \mu)/((\sigma)/(√(n)))`

In which X is the sample mean,
`\mu` is the expected mean,
`\sigma` is the standard deviation and n is the size of the sample.

Sample of 1300 voters:

This means that
`n = 1300`

Found that 45% of the residents favored construction.

This means that
`X = 0.45`

A political strategist wants to test the claim that the percentage of residents who favor construction is more than 42%.

This means that
`\mu = 0.42`, and by the Central Limit Theorem:

`(\sigma)/(√(n)) = s = \sqrt{(0.42*0.58)/(1300)} = 0.0137`

So, the test statistic is:

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

`t = (0.45 - 0.42)/(0.0137)`

`t = 2.19`

The value of the test statistic is t = 2.19.