Question

5.20 High School and Beyond, Part I: The National Center of Education Statistics conducted a survey of high school seniors, collecting test data on reading, writing, and several other subjects. Here we examine a simple random sample of 200 students from this survey. Side-by-side box plots of reading and writing scores as well as a histogram of the differences in scores are shown below. 

(b) Create hypotheses appropriate for the following research question: is there an evident difference in the average scores of students in the reading and writing exam?

Ho: \(\mu_{\text {diff }}=0\)

Ha: \(\mu_{\text {diff }}<0\)

Ho: \(\mu_{\text {diff }}=0\)

Ha: \(\mu_{\mathrm{diff}} \neq 0\)

Ho: \(\mu_{\text {diff }}=0\)

Ha: \(\mu_{\mathrm{diff}}>0\)

(c) The average observed difference in scores is X_read-write = _______ -0.545, and the standard deviation of the differences is 8.887 points. Do these data provide convincing evidence of a difference between the average scores on the two exams? 

The T-test statistic is:_______  (please round to two decimal places) 

The p-value is: _______ (please round to four decimal places) 

The conclusion of the test is: Since our p-value is less than 0.10 and greater than 0.05, there is some evidence to support the alternative hypothesis. 

• Since our p-value is less than 0.001 there is extremely evidence to support the alternative hypothesis. 

• Since our p-value is less than 0.05 and greater than 0.01, there is fairly strong evidence to support the alternative hypothesis. 

• Since our p-value is less than 0.01 and greater than 0.001, there is very strong evidence to support the alternative hypothesis. 

• Since our p-value is greater than 0.10, there is no significant evidence to support the alternative hypothesis.

(d) What type of error might we have made? Explain what the error means in the context of the application. 

• Throwing Error 

• Type I - we concluded that there is a difference in reading and writing scores for all students, when there really isn't (we rejected a null that's actually true) 

• Type II - we failed to conclude that there is a difference in reading and writing scores among all students, when there really is (we failed to support an alternative that's actually true

• Fielding Error 

(e) Based on the results of this hypothesis test, would you expect a confidence interval for the average difference between the reading and writing scores to include 0? Explain your reasoning. 

• no, because most people will not earn an average score of 0 on either exam 

• yes, because there is almost a 0% chance that average reading and writing scores are the same 

• no, because we rejected the idea that average reading and writing scores are equal 

• yes, because the evidence was not strong enough to suggest that average reading and writing scores differ

Answer 1

b)

Null and Alternative hypothesis:

Ho : µ_diff = 0

Ha : µ_diff ≠ 0

c)

Test statistic:

t = (x̅d)/(sd/√n) = (-0.545)/(8.887/√200) = -0.87

df = n-1 = 199

p-value = T.DIST.2T(ABS(-0.87), 199) = 0.3868

Since our p-value is greater than 0.10, there is no significant evidence to support the alternative hypothesis.

d)

Type II - we failed to conclude that there is a difference in reading and writing scores among all students, when there really is (we failed to support an alternative that's actually true)

e)

yes, because the evidence was not strong enough to suggest that average reading and writing scores differ