In Exercises 11–14, test the claim about the difference betwee...

Question

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.
Claim: μ1<μ2; α=0.05
Population statistics:σ1=75 and σ2=105
Sample Statistics: x̅1=2435, n1=35, x̅2=2432, n2=90

Claim: μ1<μ2; α=0.05

Population statistics:σ1=75 and σ2=105

Sample Statistics: x̅1=2435, n1=35, x̅2=2432, n2=90

Accepted Answer

Irene, hello everyone. So, this question says, a researcher wants to test the claim that muse of 1 is less than muse of 2 at the alpha equals 0.05 significance level. The population standard deviations are sigma 1 equals 60 and sigma 2 equals 95. The sample means and sizes are X1 equals 1800, and 1 equals 40, and X 2 equals. 1,805 and 2 equals 85. What is the result of the hypothesis test? And here we've got 3 different answer choices labeled A through C. So first, what are the hypotheses that we're working with? Well, first, we have the null hypothesis or H not, which would state in this case that U1 is greater than or equal to mu 2. This is because the alternative hypothesis, H1. Is the claim that's being tested. So it would be that mu 1 is less than m2. From here we compute the test statistic. Using the formula in this case for the 2 samples Z test, since we're given two independent samples and the thinner deviations of both populations are known. So, the test statistic, in this case Z is equal to. X bar 1 subtracted by X 2. And divided by The square root of. Sigma 1 squared divided by N1. Added to sigma 2 squared divided by N2. And now we plug in the information that we have. So Z is equal to 1800, subtracted by. 1,805. Divided by The square root of That's a square root of 60 squared. Divided by 40 Added to 95 squared divided by 85. And this equals approximately 0.36. So now on to the critical value. Critical value and rejection region. So At alpha equals 0.05 for a left tailed test, the critical value. Is equal to -1.645. Which leads us now to the decision we have to make. Now, we compare the test statistic that we just calculated to the critical value that we determined. Or that we found, right? So here we can see that our computed C value that's -0.36 is actually greater than the critical value. And because this is true, we fail to reject. The null hypothesis. So now what does that mean for our claim and the conclusion we can make? Well, because we have failed to reject the null hypothesis, at the 0.05 significance level, there is not enough evidence to support the claim that mu one is genuinely less than mu 2. Which brings us now to our correct answer. And that happens to be option A in the multiple choice. Which reads, at the 0.05 level of significance, there is no sufficient evidence to support the claim that mu1 is less than mu 2. And there you have it. So with that being said, thank you so very much for watching and I hope you found this helpful.

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Key Concepts

Hypothesis Testing

Significance Level (α)

Two-Sample Z-Test

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