Formulate and test a hypothesis to determine if statistical evidence suggests that the graduation rate for either top liberal arts colleges or research universities in the sample Colleges and Universities exceeds 90%. Do the data support a conclusion that the graduation rate exceeds 85%? Would your conclusions change if the level of significance were 0.01 instead of 0.05?
What is the sample data?
Formulate and test a hypothesis to determine if statistical evidence suggests that
the graduation rate for either top liberal arts colleges or research universities in the
sample Colleges and Universities exceeds 90%.
To formulate and test a hypothesis regarding the graduation rates for top liberal arts colleges or research universities in the sample, we can follow these steps:
Step 1: Establish the null and alternative hypotheses:
- Null hypothesis (H₀): The graduation rate for either top liberal arts colleges or research universities in the sample does not exceed 90%.
- Alternative hypothesis (H₁): The graduation rate for either top liberal arts colleges or research universities in the sample exceeds 90%.
Step 2: Gather relevant data:
Collect data on the graduation rates of the top liberal arts colleges and research universities in the sample. Ensure that the data includes information on both the institutions' categorization and their respective graduation rates.
Step 3: Perform statistical analysis:
Calculate the average graduation rates and determine if they exceed 90%. Conduct a hypothesis test using appropriate statistical methods, such as a t-test or z-test, depending on the characteristics of the data.
Step 4: Establish the significance level:
Specify the desired level of significance before conducting the hypothesis test. In this case, let's consider a significance level of 0.05.
Step 5: Determine the critical value:
Consult a statistical table or use statistical software to find the critical value corresponding to a significance level of 0.05. This critical value is the threshold beyond which we reject the null hypothesis.
Step 6: Compare the test statistic to the critical value:
Calculate the test statistic using the collected data and compare it to the critical value. If the test statistic is greater than the critical value, we reject the null hypothesis.
Step 7: Analyze the results:
If the null hypothesis is rejected, we can conclude that statistical evidence suggests the graduation rate for either top liberal arts colleges or research universities in the sample exceeds 90%. Conversely, if the null hypothesis is not rejected, we cannot conclude that the graduation rate exceeds 90%.
Now, to determine if the data support a conclusion that the graduation rate exceeds 85%, we would follow the same steps, but modify the null and alternative hypotheses accordingly:
- Null hypothesis (H₀): The graduation rate for either top liberal arts colleges or research universities in the sample does not exceed 85%.
- Alternative hypothesis (H₁): The graduation rate for either top liberal arts colleges or research universities in the sample exceeds 85%.
By performing the same hypothesis test and comparing the test statistic to the critical value, we can determine if the data provide evidence to support the conclusion that the graduation rate exceeds 85%.
However, in the last part of your question, you asked if the conclusions would change if the level of significance was 0.01 instead of 0.05. Changing the significance level alters the threshold of evidence required to reject the null hypothesis.
If the level of significance is reduced to 0.01, the critical value would be more stringent, and the test statistic would need to be even greater to reject the null hypothesis. Therefore, the conclusions might change depending on the new critical value and the results of the hypothesis test. A higher degree of evidence would be required to reject the null hypothesis, making it less likely to conclude that the graduation rate exceeds the given threshold.