(a) To compute the test statistic, we'll use the formula:
t = (x̄ - µ) / (s / √n)
Given x̄ = 76.9, µ = 80, s = 8.5, and n = 22, we can substitute these values into the formula:
t = (76.9 - 80) / (8.5 / √22)
Calculating this expression, we get:
t ≈ -1.982
Therefore, the test statistic is approximately -1.982.
(b) To determine the critical value, we need to find the Z-score that corresponds to a right area of 0.02. Using the inverse normal distribution function, we can calculate this:
Z_α = invNorm(1 - α, µ, σ)
Given α = 0.02, we can substitute this value into the formula:
Z_0.02 = invNorm(1 - 0.02, 0, 1)
Calculating this expression, we find:
Z_0.02 ≈ -2.054
Therefore, the critical value is approximately -2.054.
(c) Since we are dealing with a t-test, we need to use a t-distribution to depict the critical region. The critical region consists of the extreme values where we reject the null hypothesis. In this case, we reject H0 if the test statistic t falls outside the critical region.
(d) To determine whether the researcher will reject the null hypothesis or not, we compare the test statistic t with the critical value.
If t < -2.054 (the critical value), we would reject the null hypothesis. Otherwise, if t ≥ -2.054, we would fail to reject the null hypothesis.
In this case, since the test statistic t ≈ -1.982, which is greater than -2.054, the researcher would fail to reject the null hypothesis.
Therefore, the researcher will not reject the null hypothesis because the test statistic does not fall in the critical region.