Answer
Given below.
Work Step by Step
The study is to check whether there is dissimilarity between the median salary of black and white school teachers in the year 1887. The concept of Wilcoxon matched-pairs signed-ranks test is used to check the claim.
The hypotheses are formulated as
\[\begin{align}
& {{H}_{0}}:\ {{M}_{D}}=0 \\
& {{H}_{1}}:\ {{M}_{D}}\ne 0
\end{align}\]
where \[{{M}_{D}}\]denotes the median value of the differences of the salary of teachers.
To obtain the result of the test, the steps given below are followed in the Minitab software.
Step 1: Enter the data in Minitab worksheet.
Step 2: Select the “calculator” option under the Calc menu. Specify the column where the result of difference will be stored and specify the function for computing the difference in the “Expression” tab.
Step 3: Go to Stat and select 1-Sample Wilcoxon under non-parametric option.
Step 4: Input the column number of difference in the cell named “Variables.”
Step 5: Click “Test Median” and enter “0” in the cell.
Step 6: Under the option “Alternative,” select “Not equal.” Click Ok.
The output displays that the p- value is obtained as \[p=0.076\].
Since the calculated \[p=0.076\] is more than the significance level \[\alpha =0.05\], the null hypothesis is not discarded.
It can be decided that there is adequate indication to say that the median salary of two races is not dissimilar in the year 1887.
The study is to check whether there is dissimilarity between the median salary of black and white school teachers in the year 1915. The concept of Wilcoxon matched-pairs signed-ranks test is used to check the claim.
The hypotheses are formulated as
\[\begin{align}
& {{H}_{0}}:{{M}_{D}}=0 \\
& {{H}_{1}}:{{M}_{D}}\ne 0
\end{align}\]
where \[{{M}_{D}}\] represents the median value of the differences of the salary of teachers.
To achieve the result of the test, the steps given below are followed in the Minitab software.
Step 1: Enter the data in Minitab worksheet.
Step 2: Select the “calculator” option under the Calc menu. Specify the column where the result of difference will be stored and specify the function for computing the difference in the “Expression” tab.
Step 3: Go to Stat and select 1-Sample Wilcoxon under nonparametric option.
Step 4: Input the column number of difference in the cell named “Variables.”
Step 5: Click “Test Median” and enter “0” in the cell.
Step 6: Under the option “Alternative,” select “Not equal.” Click Ok.
The output shows that the p- value is obtained as \[p=0.000\].
Since the calculated \[p=0.000\] is less than the significance level \[\alpha =0.05\], the null hypothesis is discarded.
It can be decided that there is adequate indication to say that median salary of two races is dissimilar in the year 1915.
The school children and the teachers are chosen from the same countries for both years 1887 and 1915. Hence, the samples are dependent samples. Therefore, paired t-test can be used to analyze the data instead of Wilcoxon matched-pairs signed-ranks test, which is used for the dependent samples.
Section 1:
The study is to check whether there is dissimilarity between the mean length of school years of black and white children in the year 1887. The concept of paired t-test is used to check the claim.
The hypotheses are formulated as
\[\begin{align}
& {{H}_{_{0}}}:{{\mu }_{d}}=0 \\
& {{H}_{1}}:{{\mu }_{d}}\ne 0 \\
\end{align}\]
where \[{{\mu }_{d}}\] represents the mean of the differences of the length of school years.
To obtain the result of the test, the steps given below are followed in the Minitab software.
Step 1: Enter the data in Minitab worksheet.
Step 2: Choose the “Basic Statistics” option under the “Stat” menu and select “Paired t.”
Step 3: Click on “Samples in different columns” and enter the variables in the box named “First sample” and “Second Sample,” respectively.
Step 4: Specify the value of confidence level, test difference, and alternative with 95, 0, and not equal, respectively.
Step 5: Click Ok two times.
Since the calculated \[p=0.000\] is less than the significance level \[\alpha =0.05\], the null hypothesis is discarded.
It can be decided that there is adequate indication to say that the mean length of the school years is dissimilar for white and black children in the year 1887.
Section 2:
The study is to check whether there is dissimilarity between the mean length of school years of black and white children in the year 1915. The concept of paired t-test is used to check the claim.
The hypotheses are formulated as
\[\begin{align}
& {{H}_{_{0}}}:{{\mu }_{d}}=0 \\
& {{H}_{1}}:{{\mu }_{d}}\ne 0 \\
\end{align}\]
where \[{{\mu }_{d}}\] represents the mean of the differences of the length of school years.
To obtain the result of the test, the steps given below are followed in the Minitab software.
Step 1: Enter the data in Minitab worksheet.
Step 2: Choose the “Basic Statistics” option under the “Stat” menu and select “Paired t.”
Step 3: Click on “Samples in different columns” and enter the variables in the box named “First sample” and “Second Sample,” respectively.
Step 4: Specify the value of confidence level, test difference, and alternative with 95, 0, and not equal, respectively.
Step 5: Click Ok two times.
Since the calculated \[p=0.000\] is less than the significance level \[\alpha =0.05\], the null hypothesis is discarded.
It can be decided that there is adequate indication to say that the mean length of the school years is dissimilar for white and black children in the year 1915.
Section 3:
The study is to check whether there is dissimilarity between the mean salary of black and white school teachers in the year 1887. The concept of paired t-test is used to check the claim.
The hypotheses are formulated as
\[\begin{align}
& {{H}_{_{0}}}:{{\mu }_{d}}=0 \\
& {{H}_{1}}:{{\mu }_{d}}\ne 0 \\
\end{align}\]
where \[{{\mu }_{d}}\] represents the mean of the differences of the salary.
To get the result of the test, the steps given below are followed in the Minitab software.
Step 1: Enter the data in Minitab worksheet.
Step 2: Choose the “Basic Statistics” option under the “Stat” menu and select “Paired t.”
Step 3: Click on “Samples in different columns” and enter the variables in the box named “First sample” and “Second Sample,” respectively.
Step 4: Specify the value of confidence level, test difference, and alternative with 95, 0, and not equal, respectively.
Step 5: Click Ok two times.
The output shows that the p- value is obtained as \[p=0.060\].
Since the calculated \[p=0.060\] is more than the significance level \[\alpha =0.05\], the null hypothesis is not discarded.
It can be decided that there is adequate indication to say that the mean salary of two races is not dissimilar in the year 1887.
Section 4:
The study is to check whether there is dissimilarity between the mean salary of black and white school teachers in the year 1915. The concept of paired t-test is used to check the claim.
The hypotheses are formulated as
\[\begin{align}
& {{H}_{_{0}}}:{{\mu }_{d}}=0 \\
& {{H}_{1}}:{{\mu }_{d}}\ne 0 \\
\end{align}\]
where \[{{\mu }_{d}}\]represents the mean of the differences of the salary.
To obtain the result of the test, the steps given below are followed in the Minitab software.
Step 1: Enter the data in Minitab worksheet.
Step 2: Choose the “Basic Statistics” option under the “Stat” menu and select “Paired t”.
Step 3: Click on “Samples in different columns” and enter the variables in the box named “First sample” and “Second Sample,” respectively.
Step 4: Specify the value of confidence level, test difference, and alternative with 95, 0, and not equal, respectively.
Step 5: Click Ok two times.
Since the calculated \[p=0.000\] is bigger than the significance level \[\alpha =0.05\], the null hypothesis is not discarded.
It can be decided that there is adequate indication to say that the mean salary of two races is dissimilar in the year 1915.
The assumption of dependent samples is satisfied by the dataset as the data is collected from the same country in both years. To analyze the difference of the length of school year of the children and the salary of the school teachers, the concept of Wilcoxon matched-pairs signed-ranks test and paired t-test are used.
In the Wilcoxon matched-pairs signed-ranks test, the null hypothesis describes that the median value of the difference of the length of the school year of the children as well as the salary of the school teachers is equal to zero. The alternative hypothesis indicates that the median value differs from zero.
In the paired t-test, the null hypothesis describes that the mean value of the difference of length of the school year of the children as well as the salary of the school teachers is equal to zero. The alternative hypothesis indicates that the mean value differs from zero.
As \[p=0.000\] is less than the significance level \[\alpha =0.05\], it can be said that the length of the school years is different for white and black children in the year 1887 as well as in the year 1915.
On the basis of the p-value obtained for the salary of the teachers, it can be said that the salary is not dissimilar in the year 1887, but the salary of the teachers of the two races differs in the year 1915.
