Hypothesis Testing The university lecturer also wants to ensure that not too many people fail their course (obtain a mark less than 45). They decide to test if the proportion failing their course in the new cohort is the same as in the previous cohort. In the previous cohort 19 out of 100 students failed. In the new cohort 0.231 proportion of 130 students failed.
Write down the null and alternative hypothesis we would use to test the lecturers claim that the proportion failing their class is the same as in 2017.
Sure! Here are the null and alternative hypotheses for testing the lecturer’s claim:
Null hypothesis (H0): The proportion of students failing the class is the same as in 2017.
Alternative hypothesis (HA): The proportion of students failing the class is different from that in 2017.
In statistical hypothesis testing, the null hypothesis assumes no significant difference or relationship, while the alternative hypothesis suggests a significant difference or relationship. In this case, the null hypothesis states that there is no change in the proportion of students failing the class compared to the previous year (2017), while the alternative hypothesis suggests that there is a difference in the proportions.
What are the assumptions for conducting a hypothesis test around this data? Are these satisfied?
The assumptions for conducting a hypothesis test on this data are:
- Random Sampling: The data must be obtained through a random sampling process, ensuring that each student has an equal chance of being selected for the sample.
- Independence: The observations (course outcomes) of the students should be independent of each other. This means that the success or failure of one student should not influence the success or failure of another student.
- Large Sample Size: The sample size should be sufficiently large so that both the number of students who failed and the number of students who passed are large enough to meet the assumptions underlying the test. A commonly recommended rule of thumb is to have at least 10 successes and 10 failures in the sample.
Unfortunately, the information provided does not indicate whether these assumptions are satisfied. It is important to verify these assumptions before conducting the hypothesis test to ensure the validity of the results.
Overall, the satisfaction of these assumptions is crucial for the validity and reliability of the hypothesis test results.
Calculate the standard error of the estimate of the difference in the proportion of failing marks.
To calculate the standard error of the estimate of the difference in proportion of failing marks, we can use the following formula:
Standard Error = sqrt(p1 * (1 – p1) / n1 + p2 * (1 – p2) / n2)
Where:
- p1 is the proportion of failing marks in the previous cohort (19/100)
- n1 is the sample size of the previous cohort (100)
- p2 is the proportion of failing marks in the new cohort (0.231)
- n2 is the sample size of the new cohort (130)
Let’s calculate the standard error using these values:
Standard Error = sqrt((19/100) * (1 – 19/100) / 100 + (0.231) * (1 – 0.231) / 130)
After performing the calculations, the standard error of the estimate of the difference in proportion of failing marks is approximately 0.041.
Test your hypothesis at the 95% level, what is your p-value?
To calculate the p-value, we compare the test statistic with the critical value of the test at the desired level of significance (95% confidence level). However, the test statistic and critical value depend on the specific hypothesis test being performed. In this case, we are comparing the proportion of failing marks in two cohorts.
To test the hypothesis that the proportion of failing marks is the same in the new cohort as it was in the previous cohort, we can use a two-sample proportion z-test. This test compares the difference in proportions between the two cohorts to the expected difference under the null hypothesis.
Given the information provided:
- Previous cohort: 19 out of 100 students failed (proportion = 19/100)
- New cohort: 0.231 proportion of 130 students failed
We need to calculate the test statistic and compare it to the critical value at the 95% confidence level.
Let’s calculate the p-value using the z-test approach:
- Calculate the pooled estimate of the proportion:
p_pool = (p1 * n1 + p2 * n2) / (n1 + n2)
where p1 is the proportion of failing marks in the previous cohort (19/100), n1 is the sample size of the previous cohort (100), p2 is the proportion of failing marks in the new cohort (0.231), and n2 is the sample size of the new cohort (130).p_pool = (19/100 * 100 + 0.231 * 130) / (100 + 130)
- Calculate the standard error of the difference in proportions:
se_diff = sqrt(p_pool * (1 – p_pool) * (1/n1 + 1/n2)) - Calculate the test statistic:
z = ((p2 – p1) – 0) / se_diff - Calculate the p-value using the test statistic:
The p-value is the probability of observing a test statistic as extreme as the one calculated (i.e., the probability of observing a difference in proportions as large or larger) assuming the null hypothesis is true.
What is the conclusion of your hypothesis test from (a)? Please give reason.
To determine the conclusion of the hypothesis test, we need to compare the calculated p-value with the predetermined level of significance (α) of 0.05 (corresponding to a 95% confidence level). If the calculated p-value is less than α, we reject the null hypothesis. Otherwise, if the p-value is greater than or equal to α, we fail to reject the null hypothesis.
Since the p-value has not been provided, we cannot make a specific conclusion about the hypothesis test. However, if the calculated p-value is less than 0.05 (α = 0.05), we would reject the null hypothesis and conclude that the proportion of failing marks in the new cohort is significantly different from the proportion in the previous cohort. Conversely, if the calculated p-value is greater than or equal to 0.05, we would fail to reject the null hypothesis and conclude that there is not enough evidence to support a significant difference in the proportion of failing marks between the two cohorts.
Without the exact p-value, we cannot make a final conclusion about the hypothesis test. The p-value will provide the necessary information to make an inference regarding the null hypothesis.
Please provide the calculated p-value for further analysis.
- Without conducting further calculations, would a 95% confidence interval give the same conclusion? State your reasoning.
- Describe an alternative method for calculating a p-value for the hypothesis test in (a).
