Null Hypothesis

Question

Researchersareinterestedinthemeanageofacertainpopulation.Arandomsampleof10individuals ̄

drawn from the population of interest has a mean of X = 27. Assuming that the population is approximately normally distributed with variance σ = 20, can we conclude that the mean is different from μ = 30 years ? (α = 95%) .

Answer 1

There is a mistake in the way this problem is written. σ is the symbol for standard deviation, not variance. If the variance is in fact 20, σ is √20 ≈ 4.47

H₀: μ = 30

H_a: μ ≠ 30

Since the population variance is known, it is appropriate to use the normal distribution to test the hypothesis. We build a "do not reject" region as 30 ± 1.96 (4.47/√10) = 30 ± 2.77.

The sample mean of 27 falls outside the "do not reject" interval (27.23, 32.77), so we jejhect the null hypothesis and conclude that the mean is not 30, at the 5% significance level.

Answer 2

H_o: μ = 30 years

H_a: μ ≠ 30 years

z = (27 - 30)/(20/√10) = -0.47

P(z ≤ -0.47) = 0.3192 > 0.05 = α

Since the p-value is greater than the level of significance, we fail to reject the null hypothesis. There is no sufficient evidence to show that the population mean for a certain population is different.