Question

A sample of size 12, taken from a normally distributed population has a sample mean of 85.56 and a sample standard deviation of 9.70. Suppose that we have adopted the null hypothesis that the actual population mean is equal to 89, that is, H0 is that μ = 89 and we want to test the alternative hypothesis, H1, that μ ≠ 89, with level of significance α = 0.1.

a) What type of test would be appropriate in this situation?

i) A right-tailed test. ii) A left-tailed test. iii) A two-tailed test iv) None of the above.

b) What is the critical value? (for a two-tailed test give the positive value). For full marks your answer should be accurate to at least two decimal places.

c) What is the computed test statistic? For full marks your answer should be accurate to at least two decimal places.

d) Based on your test statistic and the decision rule you have decided upon, what can we conclude about H₀?

i) There is sufficient evidence, at the given significance level, to reject H₀. ii) There is insufficient evidence, at the given significance level, to reject H₀; or we fail to reject H₀. iii) There is insufficient evidence to make it clear as to whether we should reject or not reject the null hypothesis

Answer 1

Given that, sample size (n) = 12 sample mean = 85.56 and

sample standard deviation (s) = 9.70

The null and alternative hypotheses are,

H0 : μ = 80

H1 : μ ≠ 89

a) This hypothesis test is two-tailed test.

b) Degrees of freedom = 12 - 1 = 11

t-critical values at significance level of 0.1 with 11 degrees of freedom are, tcrit = ± 1.796

=> Critical value = 1.796

c) Test statistic is,

=> Test statistic = t = -1.229

d) Decision Rule : Reject H0, if t < -1.796 OR t > 1.796

Since, test statistic = -1.229 > -1.796, we fail to reject H0.

Conclusion : There is insufficient evidence, at the given significance level; we fail to reject H0.