Consider the following hypothesis test. The following data are from matched samples taken from two populations. Population Element 1 2 1 21 20 2 28 26 3 18 18 4 20 20 5 26 24 Compute the difference value for each element. If your answer is "0", enter 0.a. Compose the difference valuefor each element.b. Compute d.c.Compute the standard deviation sd.d.Conduct a hypothesis test using α=.05.What is yourconclusion?

Answer:

a) d: 1  2  0  0  2

b)

c)

d)

So the p value is lower than the significance level provided, so then we can conclude that we reject the null hypothesis that the difference mean between population 1 and 2 is less than 0. And we can say that the mean difference is higher than 0 at 5% of significance.  

Step-by-step explanation:

A paired t-test is used to compare two population means where you have two samples in  which observations in one sample can be paired with observations in the other sample. For example  if we have Before-and-after observations (This problem) we can use it.  

Let put some notation  

x=Population 1 , y = population 2

x: 21  28  18  20  26

y: 20 26  18  20  24  

a. Compose the difference valuefor each element

Let d =x-y, so the values for the difference are:

d: 1  2  0  0  2

b. Compute d.

We need the mean for the difference. If we use the following formula we got:

c.Compute the standard deviation sd.

For the standard deviation we can use the following formula:

d.Conduct a hypothesis test using α=.05.What is yourconclusion?

If we assume that the system of hypothesis for this case are:

Null hypothesis:

Alternative hypothesis:

We can calculate the statistic given by :

The next step is calculate the degrees of freedom given by:

Now we can calculate the p value, since we have a right tailed test the p value is given by:

So the p value is lower than the significance level provided, so then we can conclude that we reject the null hypothesis that the difference mean between population 1 and 2 is less than 0. And we can say that the mean difference is higher than 0 at 5% of significance.  

Final answer:

To complete this hypothesis test, calculate the difference value for each element, compute the sample mean difference (d), calculate the standard deviation (sd), and then conduct the hypothesis test using a significance level of α = 0.05.

Explanation:

To compute the difference value for each element, subtract the second value from the first value for each pair. For example, the difference for the first pair is 1-2 = -1. Repeat this for each pair of elements in the data set.

To compute the sample mean difference (d), add up all the difference values and divide by the total number of pairs. In this case, d = (-1 + 1 + 2 + 0 + 2)/5 = 0.8.

To compute the standard deviation (sd), first calculate the squared difference value for each pair and sum them up. Then divide the sum by (n-1), where n is the total number of pairs. Finally, take the square root of the result. In this case, sd = sqrt(((1-0.8)^2 + (2-0.8)^2 + (0-0.8)^2 + (2-0.8)^2)/4) = 1.32.

For the hypothesis test, we compare the sample mean difference (d) to the population mean difference (μd) under the null hypothesis. The null hypothesis states that there is no difference between the two populations. If the difference between d and μd is statistically significant at a significance level of α = 0.05, we reject the null hypothesis and conclude that there is a significant difference between the two populations. Otherwise, we fail to reject the null hypothesis.

Without information about the population mean difference (μd), we cannot perform the hypothesis test or draw a conclusion.

Learn more about Hypothesis Testing here:

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