Question

The comparisons of Scholastic Aptitude Test (SAT) scores based on the highest level of education attained by the test taker's parents are provided. A research hypothesis was that students whose parents had attained a higher level of education would on average score higher on the SAT. The overall mean SAT math score was 514. SAT math scores for independent samples of students follow. Two samples are contained in the Excel Online file below. The first sample shows the SAT math test scores for students whose parents are college graduates with a bachelor's degree. The second sample shows the SAT math test scores for students whose parents are high school graduates but do not have a college degree.

Student’s Parents
College
485
534
650
554
550
572
497
592
​Grads
487
533
526
410
515
578
448
469
​High School
442
580
479
486
528
524
​Grads
492
478
425
485
390
535
a. Formulate the hypotheses that can be used to determine whether the sample data support the hypothesis that students show a higher population mean math score on the SAT if their parents attained a higher level of education. population mean math score parents college grads. population mean math score parents high school grads. _____ _____
b. What is the point estimate of the difference between the means for the two populations?
c. Compute the p-value for the hypothesis test.
d. At αα= .05, what is your conclusion?

Answer 1

H0 is rejected and Ha is accepted that the students show a higher population mean math score on the SAT if their parents attained a higher level of education.

Explanation:

Part a:

The null and alternate hypothesis can be formulated as

H0 : u1 ≤ u2 the two means ( of students whose parents did or did not attain a higher level of education) are equal .

against the claim

Ha: u1 > u2 the students show a higher population mean math score on the SAT if their parents attained a higher level of education.

Part b:

The point estimate of the difference between the means for the two populations is the difference of sample means

x1`- x2`

525- 487= 38

Student’s Parents

College Grads

x x²

485 487 235,225 237,169

534 533 285,156 284,089

650 526 422,500 267,676

554 410 306,916 168,100

550 515 302,500 265,225

572 578 327,184 334,084

497 448 247,009 200,704

592 469 350,464 219,961

∑xi = 8400 ∑xi ²= 4,462,962

x1`= ∑ xi/n1= 8400/16= 525

Using statistic calculator Using formula : σ(n-1)

s1= 59.4205

High School Grads

x x²

442 492 195,364 242,064

580 478 336,400 228,484

479 425 229,441 180,625

486 485 236,196 235,225

528 390 278,784 152,100

524 535 274,576 286,225

∑xi = 5844 ∑xi ²= 2,875,484

x2`= ∑ xi/n2= 5844/12= 487

Using statistic calculator Using formula : σ(n-1)

s2= 51.7476

x1`- x2`= 525- 487= 38

The test statistic is

t= (x1`- x2`) / √ s1²/n1+ s2²/n2

t= 38/ √(59.4205)²/16 + (51.7476)²/12

t=38 / √3530.7958/16 + 2677.8141/12

t= 1.804

and the degrees of freedom is

υ = [s₁²/n1 + s₂²/n2]²/ (s₁²/n1 )²/ n1-1 + (s₂²/n2)²/n2-1

= [3530.7958/16 + (2677.8141/12) ]²/ (3530.7958/16)²/15 +(2677.8141/12)²/11

≈ 25

The degrees of freedom is always rounded in this calculation

From the table t∝ = 1.708

Hence critical value is t ≥ t∝

Reject H0:

Part C.

The p-value is 0.041647.

The result is less than 0.05.

Result:

H0 is rejected and Ha is accepted that the students show a higher population mean math score on the SAT if their parents attained a higher level of education.