Question

A STAT 200 instructor wants to know if students tend to score differently on the lesson 4 and 5 quizzes. Data were collected from a representative sample of 60 students during the Summer 2018 semester. Data were paired by student. The mean difference, computed as Lesson 4 Quiz – Lesson 5 Quiz, was 0.3833 points with a standard deviation of 1.9891 points.

A. Explain why it is appropriate to use the t distribution to approximate the sampling distribution in this scenario.

B. In Minitab Express, conduct a paired means t test to determine if there is evidence that lesson 4 and 5 quiz scores are different in the population of all STAT 200 students. Use the five-step hypothesis testing procedure and remember to include all relevant Minitab Express output. Step 1: Check assumptions and write hypotheses Step 2: Calculate the test statistic Step 3: Determine the p value Step 4: Decide between the null and alternative hypotheses Step 5: State a “real world” conclusion

C. In Minitab Express, conduct a single sample mean t test given a sample size of 60, sample mean of 0.3833, and sample standard deviation of 1.9891 to determine if there is evidence that the population mean is different from 0. Use the five-step hypothesis testing procedure and remember to include all relevant Minitab Express output. Step 1: Check assumptions and write hypotheses Step 2: Calculate the test statistic Step 3: Determine the p value Step 4: Decide between the null and alternative hypotheses Step 5: State a “real world” conclusion

D. Explain why your test statistic and p value were the same in parts B and C. E. What minimum sample size would be necessary to construct a 95% confidence interval for the mean difference in lesson 4 and 5 quiz scores with a margin of error of 0.20 points?

Answer 1

Given the number of students in the sample = n = 60

The sample mean is given as and the standard deviation is

(A) We can use a t-distribution here because a t-distribution provides a good approximation to the normal distribution for significantly large samples. T distribution is a sampling distribution that can be used when the true population variance is not known. Since we do not know the true population variance of the differences hence it is appropriate to use the t distribution to approximate the sampling distribution in this scenario.

(B)  Step 1: Check assumptions and write hypotheses

The underlying assumption here is that the underlying population of the difference n scores s normally distributed.

The hypothesis to be tested here is:

H₀: There is no significant difference in the test scores achieved by the students on lesson 4 and lesson 5, that is .

Against the alternative hypothesis

H₁: There is a significant difference in the test scores achieved by the students on lesson 4 and lesson 5, that is ​.

Step 2: Calculate the test statistic

The test statistics is given as:

Putting the values we have:

Step 3: Determine the p value

The p-value is gven as:

Step 4: Decide between the null and alternative hypotheses

Since the p-value is greater than the significance level 0.05, therefore we do not have sufficient evidence to reject the null hypothesis.

Therefore, We accept H₀

Step 5: State a “real world” conclusion

Thus from the above test we conclude that There is no significant difference in the test scores achieved by the students on lesson 4 and lesson 5.

(C) Given the number of students in the sample = n = 60

The sample mean is given as and the standard deviation is

Step 1: Check assumptions and write hypotheses

The underlying assumption here is that the underlying population of the difference n scores s normally distributed.

The hypothesis to be tested here is:

H₀: The true population mean is 0, that is .

Against the alternative hypothesis

H₁: The true population mean is not 0, that is ​.

Step 2: Calculate the test statistic

The test statistics is given as:

Putting the values we have:

Step 3: Determine the p value

The p-value is gven as:

Step 4: Decide between the null and alternative hypotheses

Since the p-value is greater than the significance level 0.05, therefore we do not have sufficient evidence to reject the null hypothesis.

Therefore, We accept H₀

Step 5: State a “real world” conclusion

Thus we conclude that the populaton mean is not different from 0.

(D) The results n the two tests are identcal because we are using the same sampling distribution t₅₉ to test the hypothesis.

(E) The minimum sample size is given as: