Test the claim about the mean of the differences for a populat...

Question

Test the claim about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed.
Claim: μd<0 , α=0.05 , Sample statistics: d̄ =1.5 , sd=3.2 , n=14

Claim: μd<0 , α=0.05 , Sample statistics: d̄ =1.5 , sd=3.2 , n=14

Accepted Answer

All right, hello, everyone. So, this question says, a scientist tests the claim that the mean of the differences for paired data is greater than 0 at the alpha equals 0.05 significance level. Assume random dependent samples and normal populations. The claim is that mu sub D is greater than 0. And Alpha equals 0.05. The sample statistics are D equals 2.4, S sub D equals 2.1, and N equals 12. What is the result of this test? And here we have 4 different answer choices labeled A through D. All right, so first, what are the hypotheses that are relevant for this test? The null hypothesis, that's H0, would state that mu sub D is equal to 0. Whereas the alternative hypothesis, HA corresponds to the claim. So it's that mu sub D is greater than 0. Let's begin now with the test statistics. Now, the test statistic T is equal to D subtracted by mu sub D. Divided by S sub D divided by the square root of N. So plugging in the information that we have, T is equal to 2.4 subtracted by 0. Divided by 2.1 divided by the square root of 12. And this gives you approximately 3.96. Now that we've obtained the test statistic, we now have to find the critical value for a right tailed test. At alpha equals 0.05. But recall that we also have to know the degrees of freedom. The degrees of freedom being equal to the sample size and subtracted by 1. So here that's 12 subtracted by 1, gives you 11 degrees of freedom. So, the critical T value at alpha equals 0.05, and with 11 degrees of freedom. Is approximately 1.796. Which now brings you to the comparison, right? You now have to compare the test statistic to the critical value. When you do so, you realize that the test statistic, 3.96 is greater than 1.796, the critical value. And so because the test statistic is greater than the critical value. And this is a right-tailed test, we can therefore reject the null hypothesis. Keep in mind that the reason this is a right tailed test is because we're checking if. The mean of the differences is greater than 0. So to the right of 0 on a number line. So now that we know we can reject the null hypothesis, what does this tell you about the conclusion you can make? Well, because we've rejected the null hypothesis, there is enough evidence to support the claim that mu sub D is equal, excuse me, is greater than 0 at alpha equals 0.05. So, our correct answer is going to be option A. There is sufficient evidence to support the claim. And there you have it. So with that being said, thank you so very much for watching and I hope you found this helpful.

Test the claim about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed.
Claim: μd<0 , α=0.05 , Sample statistics: d̄ =1.5 , sd=3.2 , n=14

Key Concepts

Paired Data

Hypothesis Testing

Level of Significance (α)

Watch next

Test the claim about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed.Claim: μd<0 , α=0.05 , Sample statistics: d̄ =1.5 , sd=3.2 , n=14

Key Concepts

Paired Data

Hypothesis Testing

Level of Significance (α)

Watch next

Test the claim about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed.
Claim: μd<0 , α=0.05 , Sample statistics: d̄ =1.5 , sd=3.2 , n=14