Question

In: Statistics and Probability

A standardized exam consists of three parts: math, writing, and critical reading. Sample data showing the...

A standardized exam consists of three parts: math, writing, and critical reading. Sample data showing the math and writing scores for a sample of 12 students who took the exam follow.

Student Math Writing
1 540 474
2 432 374
3 528 463
4 574 612
5 448 414
6 502 526
7 480 430
8 499 459
9 610 615
10 572 541
11 390 335
12 593 613

(a)

Use a 0.05 level of significance and test for a difference between the population mean for the math scores and the population mean for the writing scores. (Use math score − writing score.)

Formulate the hypotheses.

H0: μd ≤ 0

Ha: μd > 0

H0: μd ≠ 0

Ha: μd = 0

     

H0: μd ≤ 0

Ha: μd = 0

H0: μd = 0

Ha: μd ≠ 0

H0: μd > 0

Ha: μd ≤ 0

Calculate the test statistic. (Round your answer to three decimal places.)

Calculate the p-value. (Round your answer to four decimal places.)

p-value =

What is your conclusion?

Reject H0. We cannot conclude that there is a significant difference between the population mean scores for the math test and the writing test. Reject H0. We can conclude that there is a significant difference between the population mean scores for the math test and the writing test.      Do not reject H0. We can conclude that there is a significant difference between the population mean scores for the math test and the writing test. Do not reject H0. We cannot conclude that there is a significant difference between the population mean scores for the math test and the writing test.

(b)

What is the point estimate of the difference between the mean scores for the two tests? (Use math score − writing score.)

What are the estimates of the population mean scores for the two tests?

Math Writing

Which test reports the higher mean score?

The math test reports a  ---Select--- higher lower mean score than the writing test.

Solutions

Expert Solution

SAMPLE 1 SAMPLE 2 difference , Di =sample1-sample2 (Di - Dbar)²
540 474 66.000 1600.000
432 374 58.000 1024.000
528 463 65.000 1521.000
574 612 -38.000 4096.000
448 414 34.000 64.000
502 526 -24.000 2500.000
480 430 50.000 576.000
499 459 40.000 196.000
610 615 -5.000 961.000
572 541 31.000 25.000
390 335 55.000 841.000
593 613 -20.000 2116.000
sample 1 sample 2 Di (Di - Dbar)²
sum = 6168 5856.00 312.000 15520.000

a)

Ho :   µd=   0
Ha :   µd ╪   0
      

mean of difference ,    D̅ =ΣDi / n =   26.000                  
                          
std dev of difference , Sd =    √ [ (Di-Dbar)²/(n-1) =    37.5621                  
                          
std error , SE = Sd / √n =    37.5621   / √   12   =   10.8432      
                          
t-statistic = (D̅ - µd)/SE = (   26   -   0   ) /    10.8432   =   2.398

Degree of freedom, DF=   n - 1 =    11  
  
p-value =        0.0354   [excel function: =t.dist.2t(t-stat,df) ]

Reject H0. We can conclude that there is a significant difference between the population mean scores for the math test and the writing test.

b)

point estimate of the difference between the mean scores for the two tests=26.00

math=514

writing = 488

The math test reports a  higher mean score than the writing test.


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