Question

In Pennsylvania the average IQ score is 101.5. The variable is normally distributed, and the population standard deviation is 15. A school superintendent claims that the students in her school district have an IQ higher than the average of 101.5. She selects a random sample of 30 students and finds the mean of the test scores is 106.4. Test the claim at ???? = 0.05.

Answer 1

We conclude that the Pennsylvania school district have an IQ higher than the average of 101.5

Explanation:

We are given the following in the question:

Population mean, μ = 101.5

Sample mean,
`\bar{x}` = 106.4

Sample size, n = 30

Alpha, α = 0.05

Population standard deviation, σ = 15

First, we design the null and the alternate hypothesis

`H_(0): \mu = 101.5\\H_A: \mu > 101.5`

We use one-tailed z test to perform this hypothesis.

Formula:

`z_(stat) = \displaystyle\frac{\bar{x} - \mu}{(\sigma)/(√(n)) }`

Putting all the values, we have

`z_(stat) = \displaystyle(106.4 - 101.5)/((15)/(√(30)) ) = 1.789`

Now,
`z_(critical) \text{ at 0.05 level of significance } = 1.96`

Since,

`z_(stat) > z_(critical)`

We reject the null hypothesis and accept the alternate hypothesis.

Thus, we conclude that the Pennsylvania school district have an IQ higher than the average of 101.5