Question

Q2) Independent-Samples t-Test (25 points total)

A social researcher is interested in understanding the effect of college education on wages. The workers in one group have earned an associate’s degree while members of the other group hold at least a bachelor’s degree. He would like to run a hypothesis test with α = .10 to see if those with a bachelor’s degree have significantly higher hourly wages than those with an associate’s degree.

Bachelor’s degree			Associate’s degree
Participant	Hourly wage		Participant	Hourly wage
1	11.25		11	10.25
2	12.25		12	11.25
3	11.60		13	9.10
4	9.80		14	10.00
5	9.40		15	9.70
6	12.60		16	11.00
7	11.80		17	10.10
8	12.70		18	9.40
9	11.90		19	9.20
10	14.10		20	10.50

1. Calculate the pooled standard deviation for the populations and then use it to calculate the standardized effect size of this test. (2 point total: 1 for pooled standard deviation, 1 for effect size. Deduct .5 if a result is wrong but the process is correct.)

Answer 1

Since we don't know the population mean or the population variance hence, we would assume the population mean for mean hourly wage for a bachelor's degree participant to be $\mu_{b}$ and the population mean for mean hourly wage for an associate's degree participant to be $\mu_{a}$ .

Now, we need to state the null and the alternate hypothesis.

  $\mu_{b} - \mu_{a} < 0$

  
п, – ра > 0

Since we don't know the variance of the population, hence we would need to make one assumption that the variance of the population for both the bachelor's degree participants and the associate's degree participants are same.

or

Ps

Now, since we don't know the variance, hence we would calculate sample variance for both the samples. And then we will calculate the pooled sample variance that would be an estimate of the population variance.

Sample variance for bachelor's degree participants

$$ = (1893 – )

OL VL'II = 72113 ='Y

s = (X - -= 1.891

Similarly Sample variance for associate's degree participants

$S_{a}^2 = \sum \frac{(X_{ai} - \overline{X_{a}})^2}{n-1}$

Χ. - ΣXαι 100.5 = 10.05 10

Så = 5 (Xai – Xa)? - = 0.529

Now, as we know the sample variance for both the samples, hence we can calculate the pooled sample variance which would be an estimate of the population variance.

The formula for the pooled sample variance is given by

$2_ (n-1)} + (m - 1) n+m - 2

Putting the values of n and m as 10 because both are 10 in the question, we would have the Poole variance to be

58 – 9 * (S8 + S3) – (84 +) 1.891 + 0.529 – 1.21 18

Hence the pooled variance is 1.21. Hence the pooled standard deviation is answer

V1.21 = 1.1

Now as we know the pooled variance, we can now calculate the t statistics for the above test.

T (Xo - Xa) - (po - pia) Sz(1/m+1/n)

In order to test the hypothesis, we would take

0 = P – Q1

T = - (11.74 – 10.05) 1.21(1/10 + 1/10) = 3.435

Now, we need to calculate the critical t value for the two-sample hypothesis test with degree of freedom = n+m-2 = 18 and alpha = 0.1. Remember, we need to calculate the critical t value for a one tailed test. T value can be calculated from the t table.

WE have

ta,n+m-2 = 1.33

NOw, since

T> tantm-2

Hence we will reject the null hypothesis that  $\mu_{b} - \mu_{a} < 0$

Hence we can say that with alpha = 0.1, the null hypothesis that bachelor's degree participants do not have significantly higher hourly wages than associate's degree participants is not true.

Thank You!!

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