Question

It is believed that the average starting salary for a 21-25 year old college grad exceeds $52,000 per year. Hence, it is desired to test H0: µ ≤ 52,000 versus H1 : µ > 52,000. Assume σ = $5,745 and a SRS of size 65 is taken. We reject H0 when X > 53,460. The significance level α is:

Answer 1

The significance level α is 0.02.

Explanation:

A hypothesis test for single mean can be performed to determine whether the average starting salary for a 21-25 year old college grad exceeds $52,000 per year.

The hypothesis is defined as follows:

H₀: The average starting salary for a 21-25 year old college grad does not exceeds $52,000 per year, i.e. µ ≤ 52,000.

Hₐ: The average starting salary for a 21-25 year old college grad exceeds $52,000 per year, i.e. µ > 52,000.

The information provided is:

σ = $5,745

n = 65

Also, if
`\bar X>\$53,460` then the null hypothesis will be rejected.

Here, we need to compute the value of significance level α, the type I error probability.

A type I error occurs when we reject a true null hypothesis (H₀).

That is:

α = P (type I error)

α = P (Rejecting H₀| H₀ is true)

`=P(\bar X>53460|\mu \leq 52000)`

`=P[(\bar X-\mu_(0))/(\sigma/√(n))>(53460-52000)/(5745/√(65))]`

`=P(Z>2.05)\\=1-P(Z<2.05)\\=1-0.97982\\=0.02018\\\approx0.02`

*Use a z-table for the probability.

Thus, the significance level α is 0.02.