Question

Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. In addition, in each case for which the results are significant, state which group (1 or 2) has the larger mean.Hypotheses: H0:μ1=μ2 vs Ha:μ1≠μ2(a) 95% confidence interval for μ1-μ2: 0.17 to 0.32Conclusion: RejectDo not rejectH0Significance level: 90%5%1%10%95%99%Group with the larger mean:
Group 1Group 2Cannot determine(b) 99% confidence interval for μ1-μ2: -2.8 to 5.8Conclusion: RejectDo not rejectH0Significance level: 90%1%99%95%5%10%Group with the larger mean:Group 1Cannot determineGroup 2(c) 90% confidence interval for μ1-μ2: -9.6 to -3.8Conclusion:RejectDo not rejectH0Significance level:10%90%95%99%1%5%Group with the larger mean:Group 2Group 1Cannot determine

Answer 1

Answer:

Step-by-step explanation:

Hello!

To use a confidence interval to decide on the statistical hypothesis several conditions should be met.

1) The hypothesis should be two-tailed (= vs. ≠)

2) The hypothesis and the confidence interval should be made for the same parameters.

If you have for example a hypothesis for "μ₁ - μ₂", the interval should estimate "μ₁ - μ₂", if you have a calculated interval estimating "μ₂ - μ₁" it is not useful to decide over "μ₁ - μ₂".

3) The significance and confidence level should be complementary, this means if you constructed the interval with a 95% level the hypothesis test should be made at a 5% level.

4) If the CI contains the value of the parameter stated in the null hypothesis, the decision is to not reject it. If the CI does not contain the value of the parameter under the null hypothesis, the decision is to reject it.

Keep in mind that the hypothesis is two-tailed if you reject them the only clear conclusion to be made is that the means are different if one is bigger than the other cannot be determined with this test. A one-tailed test is required for that.

With this in mind, you have the hypothesis pair:

H₀: μ₁ = μ₂

H₁: μ₁ ≠ μ₂

a.

95% CI [0.17;0.32]

The interval does not contain 0, so at a significance level of 5%, the decision is to reject the null hypothesis.

The only determination that can be made is that the means of both groups are different. Since both bonds are positive, it looks like the mean of group 1 is greater than the means of group 2, to be able to determine whether this is true or not a one-tailed test with hypothesis μ₁ > μ₂ should be made.

b.

99% CI [-2.8;5.8]

The interval contains 0, so at a significance level of 1% the decision is to not reject the null hypothesis, the means of both groups are equal.

c.

90% CI [-9.6;-3.8]

The interval does not contain 0, the decision is to reject the null hypothesis. Like item a. the only determination that can be made is that both means are different.

In this case, both interval bonds are negative so it seems like the mean of group 2 greater than the mean of group one. Again, this cannot be determined by looking at the interval, you need to make a one-tailed test with hypothesis μ₁ < μ₂ to determine it.

I hope it helps!