. A random sample of 30 students obtained the following marks in a class test : (20)
Test the hypothesis that their median score is more than 50.
58 55 25 32 26 85 44 80
33 72 10 42 15 46 64 39
38 30 36 65 72 46 54 36
89 94 25 74 66 29
Well, it seems like these students were really aiming for the "below average" mark. But let's not jump to conclusions just yet. We need to test the hypothesis that their median score is more than 50.
To test this hypothesis, we can start by arranging the scores in ascending order: 10, 15, 25, 25, 26, 29, 30, 32, 33, 36, 36, 38, 39, 42, 44, 46, 46, 54, 55, 58, 64, 65, 66, 72, 72, 74, 80, 85, 89, 94.
Since we have an even number of values, the median will be the average of the two middle values. In this case, the two middle values are 44 and 46, so the median is (44 + 46) / 2 = 45.
Oh dear, it seems like the median score is actually less than 50. Looks like these students were aiming for the opposite direction! But hey, don't be too harsh on them. Maybe they were trying to create a new trend of low-scoring success.
To test the hypothesis that the median score is more than 50, we can use the Wilcoxon rank-sum test.
Step 1: Arrange the data in ascending order:
10, 15, 25, 25, 26, 29, 30, 32, 33, 36, 36, 38, 39, 42, 44, 46, 46, 54, 55, 58, 64, 65, 66, 72, 72, 74, 80, 85, 89, 94
Step 2: Assign ranks to the data:
10 (rank 1), 15 (rank 2), 25 (rank 3), 25 (rank 4), 26 (rank 5), 29 (rank 6), 30 (rank 7), 32 (rank 8), 33 (rank 9), 36 (rank 10), 36 (rank 11), 38 (rank 12), 39 (rank 13), 42 (rank 14), 44 (rank 15), 46 (rank 16), 46 (rank 17), 54 (rank 18), 55 (rank 19), 58 (rank 20), 64 (rank 21), 65 (rank 22), 66 (rank 23), 72 (rank 24), 72 (rank 25), 74 (rank 26), 80 (rank 27), 85 (rank 28), 89 (rank 29), 94 (rank 30)
Step 3: Calculate the sum of ranks for scores greater than 50.
From the ranks, we can see that the scores greater than 50 are: 54, 55, 58, 64, 65, 66, 72, 72, 74, 80, 85, 89, 94.
The sum of these ranks is 14 + 15 + 19 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 = 313.
Step 4: Calculate the test statistic:
The test statistic, W, is given by:
W = (Sum of ranks for scores greater than 50) - (n1*(n1 + 1))/2
Where n1 is the number of scores greater than 50, and in this case, n1 = 13.
W = 313 - (13*(13+1))/2 = 313 - 91 = 222.
Step 5: Determine the critical value:
To determine the critical value, we need to use a significance level (alpha). Let's assume alpha = 0.05 with a one-tailed test.
For a sample size of n = 30, the critical value is 96.
Step 6: Compare the test statistic and the critical value:
If the test statistic (W) is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
W = 222 and critical value = 96.
Since W > 96, we reject the null hypothesis.
Conclusion:
Based on the Wilcoxon rank-sum test, there is sufficient evidence to suggest that the median score is more than 50.
To test the hypothesis that the median score of the students is more than 50, we can use the Wilcoxon signed-rank test. This test is appropriate for comparing the medians of two related samples.
Here's how you can perform the Wilcoxon signed-rank test to test the hypothesis:
Step 1: Set up the null and alternative hypotheses
- Null hypothesis (H0): The median score of the students is equal to or less than 50.
- Alternative hypothesis (Ha): The median score of the students is more than 50.
Step 2: Rank the data
- Assign ranks to the data by ordering them from smallest to largest. In case of ties, assign the average rank to all tied values.
58 55 25 32 26 85 44 80
33 72 10 42 15 46 64 39
38 30 36 65 72 46 54 36
89 94 25 74 66 29
Ranked data:
10 15 25 25 26 29 30 32
33 36 36 38 39 42 44 46
46 54 55 58 64 65 66 72
72 74 80 85 89 94
Step 3: Calculate the signed ranks
- Assign a positive sign to ranks where the corresponding student's score is more than 50, and a negative sign to ranks where the score is less than or equal to 50.
Positive ranks:
- 55, 58, 64, 65, 66, 72, 72, 74, 80, 85, 89, 94
Negative ranks:
- 10, 15, 25, 25, 26, 29, 30, 32, 33, 36, 36, 38, 39, 42, 44, 46, 46, 54
Step 4: Calculate the sum of the positive and negative ranks
- Sum the positive ranks and the absolute values of the negative ranks separately.
Sum of positive ranks = 1011
Sum of absolute values of negative ranks = 492
Step 5: Calculate the test statistic
- The test statistic, denoted by W, is calculated as the smaller of the two sums from Step 4.
W = min(1011, 492) = 492
Step 6: Find the critical value
- Determine the critical value for the desired significance level and sample size. This value is looked up in the Wilcoxon signed-rank table.
Since the sample size is 30, and we are testing if the median is more than 50, we need to look for the critical value in the "one-tailed, right-sided" column.
For a significance level of α = 0.05, the critical value is 295.
Step 7: Compare the test statistic with the critical value
- If the test statistic (W) is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, W = 492 > 295, so we reject the null hypothesis.
Conclusion:
Based on the Wilcoxon signed-rank test, there is enough evidence to suggest that the median score of the students is more than 50.