Question

The College Board provided comparisons of Scholastic Aptitude Test (SAT) scores based on the highest level of education attained by the test taker's parents. 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. 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.

College Grads

485	487
550	517
634	542
538	426
566	499
588	594
513	432
592	453

High School Grads

442	492
580	478
479	425
486	485
528	390
524	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. (Let μ₁ = population mean verbal score of students whose parents are college graduates with a bachelor's degree and μ₂ = population mean verbal score of students whose parents are high school graduates but do not have a college degree.)

H₀: μ₁ − μ₂ ≤ 0

H_a: μ₁ − μ₂ > 0

H₀: μ₁ − μ₂ ≠ 0

H_a: μ₁ − μ₂ = 0

     

H₀: μ₁ − μ₂ ≥ 0

H_a: μ₁ − μ₂ < 0

H₀: μ₁ − μ₂ = 0

H_a: μ₁ − μ₂ ≠ 0

H₀: μ₁ − μ₂ < 0

H_a: μ₁ − μ₂ = 0

(b)

What is the point estimate of the difference between the means for the two populations?

(c)

Find the value of the test statistic. (Round your answer to three decimal places.)

Compute the p-value for the hypothesis test. (Round your answer to four decimal places.)

p-value =

(d)

At

α = 0.05,

what is your conclusion?

Do not Reject H₀. There is sufficient evidence to conclude that higher population mean verbal scores are associated with students whose parents are college graduates. Reject H₀. There is insufficient evidence to conclude that higher population mean verbal scores are associated with students whose parents are college graduates.     Reject H₀. There is sufficient evidence to conclude that higher population mean verbal scores are associated with students whose parents are college graduates.Do not reject H₀. There is insufficient evidence to conclude that higher population mean verbal scores are associated with students whose parents are college graduates.

Answer 1

a)

H₀: μ₁ − μ₂ ≤ 0

H_a: μ₁ − μ₂ > 0

b)

Level of Significance ,    α =    0.05              
                      
Sample #1   ---->   1              
mean of sample 1,    x̅1=   526.000              
standard deviation of sample 1,   s1 =    60.7783              
size of sample 1,    n1=   16              
                      
Sample #2   ---->   2              
mean of sample 2,    x̅2=   487.000              
standard deviation of sample 2,   s2 =    51.7476              
size of sample 2,    n2=   12              
                      
difference in sample means =    x̅1-x̅2 =    526.0000   -   487.0   =   39.000

c)

                          
pooled std dev , Sp=   √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) =    57.1321                  
std error , SE =    Sp*√(1/n1+1/n2) =    21.8177                  
                          
t-statistic = ((x̅1-x̅2)-µd)/SE = (   39.0000   -   0   ) /    21.82   =   1.788

Degree of freedom, DF=   n1+n2-2 =    26
p-value =        0.0428

d)

Reject H₀. There is sufficient evidence to conclude that higher population mean verbal scores are associated with students whose parents are college graduates.