Question

2. Consider the following data sample data obtained in a study of relationship between the number of years that applicants for a certain foreign service jobs have studied German in high school or college and the grades that they received on a proficiency test in that language tof Grade in test (V) Some calculations: 58, 48 1 5 57, 63, 7578, 72, 73 89 B9. 84 J: ΣΧ2-133, Σ. 35, ΣΥι 50085 and Σ's 2554 697 , a) Fit a least squares line to the data. b) Find the residual for the 9th applicant's data. e) Based on the data, error standard deviation is found to be 5.65. Test the hypothesis that each additional year in German in high school or college ads another 12.5 point to the expected proficiency score of an applicant. Use the alternative hypothesis Bi t 12.5 and level of significance 0.05.

Answer 1

Σn	ΣX	ΣY	ΣXY	ΣX²	ΣY²
10	35.00	697.00	2554.00	133.00	50085.00

a)

sample size ,   n =   10          
here, x̅ =   3.5000       ȳ =   69.7  
                  
SSxx =    Σx² - (Σx)²/n =   10.500          
SSxy=   Σxy - (Σx*Σy)/n =   114.500          
SSyy =    Σy²-(Σy)²/n =   1504.100

       
slope ,    ß1 = SSxy/SSxx =   10.9048          
                   intercept,   ß0 = y̅-ß1* x̄ =   31.5333          
                  
so, regression line is   Ŷ =   31.5333   +   10.9048   *x

b)

9th applicant data is (5,89)

predicted y at x=5 is

Ŷ =   31.5333   +   10.9048   * 5 = 86

residual = actual-predicted = 89 - 86 = 3

c)

Ho:   ß1=   12.5
H1:   ß1╪   12.5

std error ,Se =    5.65

estimated std error of slope =Se(ß1) = Se/√Sxx =    1.7441

t stat = ( estimated slope - ß1 )   /Se(ß1) = (10.9048-12.5)/1.7441= -0.915

DF=n-2 = 10-2 = 8

α=0.05

t-critical value= ± 2.306

since, |t stat |< |critical value |, fail to reject Ho

so, test is not significant at α=0.05