According to a report on workforce diversity, about 60% of the employees in high

Question

According to a report on workforce diversity, about 60% of the employees in high-tech firms in Silicon Valley are white and about 20% are Asian (http://moneycnn.com, November 9, 2011). Women, along with blacks and Hispanics, are highly underrepresented. Just about 30% of all employees are women, with blacks and Hispanics accounting for only about 15% of the workforce. Tara Jones is a recent college graduate, working for a large high-tech firm in Silicon Valley. She wants to determine if her firm faces the same diversity as in the report. She collects gender and ethnicity information on 50 employees in her firm. A portion of the data is shown in the accompanying table.

a.

At the 5% level of significance, determine if the proportion of women in Tara’s firm is different from 30%.

b.

At the 5% level of significance, determine if the proportion of whites in Tara’s firm is more than 50%.

c. What are the null and alternative hypothesis

Gender Ethnicity Female White Male White Male Nonwhite Male White Male Nonwhite Female White Male White Male White Female White Male White Female White Male Nonwhite Male Nonwhite Male Nonwhite Female Nonwhite Male White Female White Female Nonwhite Male White Female White Female White Male Nonwhite Male White Male White Female White Male White Female Nonwhite Male White Male White Female White Male White Female White Male White Male Nonwhite Male Nonwhite Female White Male Nonwhite Female Nonwhite Female Nonwhite Male White Male Nonwhite Male White Female White Male White Female Nonwhite Male White Female Nonwhite Male White Male White Male Nonwhite

Explanation / Answer

lets first calculate the proprtion of women in the sample , we see that there are 19 females out of 50 , so the proportion is 19/50 = 0.38

a)

here

z = (p1 - p2) / SE

SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }

p = (p1 * n1 + p2 * n2) / (n1 + n2)

so (0.38*50 +0.3*50)/100 =0.34

now SE = sqrt((0.34 *(1-0.34)*(2/50)) =0.0947

now the z is (0.38-0.3)/0.0947 =0.8447 , now we check the z table as

P ( Z<0.8447 )=0.7995 , so as the p value is not less than 0.05 , hence we conclude that the proportions are not different statistically

b)

here

z = (p1 - p2) / SE

SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }

p = (p1 * n1 + p2 * n2) / (n1 + n2)

so (0.38*50 +0.5*50)/100 =0.44

now SE = sqrt((0.44 *(1-0.44)*(2/50)) =0.099

now the z is (0.38-0.5)/0.099 = -1.121 , now we check the z table (please keep them handy) as

P ( Z>1.121 )=P ( Z<1.121 )=0.8686 , so as the p value is not less than 0.05 , hence we conclude that the proportions is not more than 50 %

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