To test: the null hypothesis and show that it is rejected at the
Explanation of Solution
Given information :
Department | A | B | C | D | E | F | Total |
Accept | 601 | 370 | 322 | 269 | 147 | 46 | 1755 |
Reject | 332 | 215 | 596 | 523 | 437 | 668 | 2771 |
Total | 933 | 585 | 918 | 792 | 584 | 714 | 4526 |
Concept Involved:
In order to decide whether the presumed hypothesis for data sample stands accurate for the entire population or not we use the hypothesis testing.
The value of test statistics and the critical value identified from the table help us to decide whether to reject or do not reject null hypothesis.
The critical value from Table A.4, using degrees of freedom of
If
The values of two qualitative variables are connected and denoted in a contingency table.
This table consists of rows and column. The variables in each row and each column of the table represent a category.
The number of rows of contingency table is represented by letter ‘r’ and number of column of contingency table is represented by letter ‘c’.
The formula to find the number of degree of freedom of contingency table is
Calculation:
Finding the expected frequency for the cell corresponding to: | The expected frequency |
Number of applicants who were accepted by department A The row total is 1755, the column total is 933, and the grand total is 4526. | |
Number of applicants who were accepted by department B The row total is 1755, the column total is 585, and the grand total is 4526. | |
Number of applicants who were accepted by department C The row total is 1755, the column total is 918, and the grand total is 4526. | |
Number of applicants who were accepted by department D The row total is 1755, the column total is 792, and the grand total is 4526. | |
Number of applicants who were accepted by department E The row total is 1755, the column total is 584, and the grand total is 4526. | |
Number of applicants who were accepted by department F The row total is 1755, the column total is 714, and the grand total is 4526. | |
Number of applicants who were rejected by department A The row total is 2771, the column total is 933, and the grand total is 4526. | |
Number of applicants who were rejected by department B The row total is 2771, the column total is 585, and the grand total is 4526. | |
Number of applicants who were rejected by department C The row total is 2771, the column total is 918, and the grand total is 4526. | |
Number of applicants who were rejected by department D The row total is 2771, the column total is 792, and the grand total is 4526. | |
Number of applicants who were rejected by department E The row total is 2771, the column total is 584, and the grand total is 4526. | |
Number of applicants who were rejected by department F The row total is 2771, the column total is 714, and the grand total is 4526. |
All the expected frequencies are at least 5. From the results of previous part we have the below table:
Finding the value of the chi-square corresponding to: | |
Number of applicants who were accepted by department A The observed frequency is 601 and expected frequency is 361.78 | |
Number of applicants who were accepted by department B The observed frequency is 370 and expected frequency is 226.84 | |
Number of applicants who were accepted by department C The observed frequency is 322 and expected frequency is 355.96 | |
Number of applicants who were accepted by department D The observed frequency is 269 and expected frequency is 307.11 | |
Number of applicants who were accepted by department E The observed frequency is 147 and expected frequency is 226.45 | |
Number of applicants who were accepted by department F The observed frequency is 46 and expected frequency is 276.86 | |
Number of applicants who were rejectedby department A The observed frequency is 332 and expected frequency is 571.22 | |
Number of applicants who were rejected by department B The observed frequency is 215 and expected frequency is 358.16 | |
Number of applicants who were rejectedby department C The observed frequency is 596 and expected frequency is 562.04 | |
Number of applicants who were rejectedby department D The observed frequency is 523 and expected frequency is 484.89 | |
Number of applicants who were rejectedby department E The observed frequency is 437 and expected frequency is 357.55 | |
Number of applicants who were rejectedby department F The observed frequency is 668 and expected frequency is 437.14 |
To compute the test statistics, we use the observed frequencies and expected frequency:
Here r represents the number of rows and c represents the number of columns.
Given
Degrees of freedom | Table A.4 Critical Values for the chi-square Distribution | |||||||||
0.995 | 0.99 | 0.975 | 0.95 | 0.90 | 0.10 | 0.05 | 0.025 | 0.01 | 0.005 | |
1 | 0.000 | 0.000 | 0.001 | 0.004 | 0.016 | 2.706 | 3.841 | 5.024 | 6.635 | 7.879 |
2 | 0.010 | 0.020 | 0.051 | 0.103 | 0.211 | 4.605 | 5.991 | 7.378 | 9.210 | 10.597 |
3 | 0.072 | 0.115 | 0.216 | 0.352 | 0.584 | 6.251 | 7.815 | 9.348 | 11.345 | 12.838 |
4 | 0.207 | 0.297 | 0.484 | 0.711 | 1.064 | 7.779 | 9.488 | 11.143 | 13.277 | 14.860 |
5 | 0.412 | 0.554 | 0.831 | 1.145 | 1.610 | 9.236 | 11.070 | 12.833 | 15.086 | 16.750 |
Conclusion:
Test statistic: 778.91; Critical value: 15.086.
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Chapter 12 Solutions
Elementary Statistics 2nd Edition
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