Question

Two teams in a laboratory are using different methods of performing biological tests. Success rate of those tests depends on the size of the sample material used for the test. The data is below. In the second column you have success rate for a given size of a sample. First 10 observations describe success rates of the first team, while another 10 observations describe success rates of the second team.

Size	Success	Team 2	Team 2*Size
1	0.219438	0	0
2	0.332883	0	0
3	0.304965	0	0
4	0.406052	0	0
5	0.414167	0	0
6	0.482654	0	0
7	0.595429	0	0
8	0.550863	0	0
9	0.663525	0	0
10	0.712554	0	0
1	0.284065	1	1
2	0.246359	1	2
3	0.219077	1	3
4	0.172007	1	4
5	0.165857	1	5
6	0.096087	1	6
7	0.082457	1	7
8	0.057341	1	8
9	0.019099	1	9
10	0.021757	1	10

1. Write out the regression equation for each team.
2. Perform regression on Success vs. Size for the first team. Write out estimated regression equation.
3. Perform regression on Success vs. Size for the second team. Write out estimated regression equation.
4. Construct a scatterplot of the Success vs. Size for both teams.
5. Suppose that the random factors are the same in case of both teams. Therefore, you can gain from regressing Success vs. Size for both teams simultaneously. Write down the regression equation for both teams together adding appropriate indicator (dummy) variables.
6. Perform the regression on Output vs. Input for both plants including appropriate indicator (dummy) variables. Write out estimated regression equation.
7. What are the gains from performing regression for both teams simultaneously? How do the results in b. and c. differ from the results in f?
8. Interpret the values of the coefficients for the dummy variables.
9. Estimate the difference between the benefit of one unit increase in the size of a sample on the success rates of both teams. Provide a 95% confidence interval for this difference.
10. Which team is more efficient for small samples? Write out the statistical test for testing the hypothesis that the second team has higher success rate for the smallest size of a sample.

Answer 1

I used R software to solve this question.

R codes and output:

> d=read.table('data.csv',header=T,sep=',')

> head(d)

Size Success Team.2 Team.2.Size

1    1 0.219438      0           0

2    2 0.332883      0           0

3    3 0.304965      0           0

4    4 0.406052      0           0

5    5 0.414167      0           0

6    6 0.482654      0           0

> attach(d)

The following objects are masked from d (pos = 3):

    Size, Success, Team.2, Team.2.Size

> fit_1=lm(Success[which(Team.2==0)]~Size[which(Team.2==0)])

> summary(fit_1)

Call:

lm(formula = Success[which(Team.2 == 0)] ~ Size[which(Team.2 ==

    0)])

Residuals:

      Min        1Q    Median        3Q       Max

-0.047976 -0.024417 -0.001235 0.015226 0.048825

Coefficients:

                         Estimate Std. Error t value Pr(>|t|)   

(Intercept)              0.180965   0.023685   7.641 6.07e-05 ***

Size[which(Team.2 == 0)] 0.052234   0.003817 13.684 7.84e-07 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.03467 on 8 degrees of freedom

Multiple R-squared: 0.959,    Adjusted R-squared: 0.9539

F-statistic: 187.3 on 1 and 8 DF, p-value: 7.836e-07

> plot(Size[which(Team.2==0)],Success[which(Team.2==0)])

> abline(fit_1)

> fit_2=lm(Success[which(Team.2==1)]~Size[which(Team.2==1)])

> summary(fit_2)

Call:

lm(formula = Success[which(Team.2 == 1)] ~ Size[which(Team.2 ==

    1)])

Residuals:

       Min         1Q     Median         3Q        Max

-0.0248730 -0.0087686 -0.0000112 0.0078027 0.0244016

Coefficients:

                          Estimate Std. Error t value Pr(>|t|)   

(Intercept)               0.306367   0.010182   30.09 1.61e-09 ***

Size[which(Team.2 == 1)] -0.030901   0.001641 -18.83 6.53e-08 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0149 on 8 degrees of freedom

Multiple R-squared: 0.9779, Adjusted R-squared: 0.9752

F-statistic: 354.6 on 1 and 8 DF, p-value: 6.534e-08

> plot(Size[which(Team.2==1)],Success[which(Team.2==1)])

> abline(fit_2)

Que.a

Can you please tell, which variables are used as independent variable for estimated regression equation.

Que.b

Regression equation for first team:

Success = 0.180965 + 0.052234 size

Que.c

Regression equation for second team:

Success = 0.306367 -  0.030901 size

Que.d

Scatter plot for first team:

Scatter plot for second team: