Question

51% of students entering four-year colleges receive a degree within six years. Is this percent smaller than for students who play intramural sports? 123 of the 292 students who played intramural sports received a degree within six years. What can be concluded at the level of significance of αα = 0.01?

1. For this study, we should use Select an answer t-test for a population mean z-test for a population proportion
2. The null and alternative hypotheses would be:
   Ho: ? p μ  Select an answer > < ≥ ≠ ≤ =   (please enter a decimal)   
   H₁: ? p μ  Select an answer = ≥ > < ≠ ≤   (Please enter a decimal)

3. The test statistic ? z t  =  (please show your answer to 3 decimal places.)
4. The p-value =  (Please show your answer to 4 decimal places.)
5. The p-value is ? ≤ >  αα
6. Based on this, we should Select an answer accept fail to reject reject  the null hypothesis.
7. Thus, the final conclusion is that ...
   • The data suggest the population proportion is not significantly smaller than 51% at αα = 0.01, so there is sufficient evidence to conclude that the population proportion of students who played intramural sports who received a degree within six years is equal to 51%.
   • The data suggest the populaton proportion is significantly smaller than 51% at αα = 0.01, so there is sufficient evidence to conclude that the population proportion of students who played intramural sports who received a degree within six years is smaller than 51%
   • The data suggest the population proportion is not significantly smaller than 51% at αα = 0.01, so there is not sufficient evidence to conclude that the population proportion of students who played intramural sports who received a degree within six years is smaller than 51%.
8. Interpret the p-value in the context of the study.
   • There is a 51% chance of a Type I error.
   • There is a 0.12% chance that fewer than 51% of all students who played intramural sports graduate within six years.
   • If the sample proportion of students who played intramural sports who received a degree within six years is 42% and if another 292 such students are surveyed then there would be a 0.12% chance of concluding that fewer than 51% of all students who played intramural sports received a degree within six years.
   • If the population proportion of students who played intramural sports who received a degree within six years is 51% and if another 292 students who played intramural sports are surveyed then there would be a 0.12% chance fewer than 42% of the 292 students surveyed received a degree within six years.
9. Interpret the level of significance in the context of the study.
   • There is a 1% chance that the proportion of all students who played intramural sports who received a degree within six years is smaller than 51%.
   • If the population proportion of students who played intramural sports who received a degree within six years is 51% and if another 292 students who played intramural sports are surveyed then there would be a 1% chance that we would end up falsely concluding that the proportion of all students who played intramural sports who received a degree within six years is smaller than 51%
   • There is a 1% chance that aliens have secretly taken over the earth and have cleverly disguised themselves as the presidents of each of the countries on earth.
   • If the population proportion of students who played intramural sports who received a degree within six years is smaller than 51% and if another 292 students who played intramural sports are surveyed then there would be a 1% chance that we would end up falsely concluding that the proportion of all students who played intramural sports who received a degree within six years is equal to 51%.

Answer 1

(a)

For this study, we should use z-test for a population proportion.

(b)

We have to test for null hypothesis

against the alternative hypothesis

(a)

Our test statistics is given by

Here,

Number of observations

(b)

[Using R-code 'pnorm(-3.034313)']

(c)

Level of significance

Here, we observe that

(d)

We reject our null hypothesis if

So, we reject our null hypothesis.

(e)

Hence, based on the given data we can conclude as follows.

The data suggest the population proportion is significantly smaller than 51% at α = 0.01, so there is sufficient evidence to conclude that the population proportion of students who played intramural sports who received a degree within six years is smaller than 51%.

(f)

The interpretation of p-value is as follows.

If the sample proportion of students who played intramural sports who received a degree within six years is 42% and if another 292 such students are surveyed then there would be a 0.12% chance of concluding that fewer than 51% of all students who played intramural sports received a degree within six years.

(g)

Level of significance denotes probability of type I error. So, the interpretation of level of significance is as follows.

If the population proportion of students who played intramural sports who received a degree within six years is 51% and if another 292 students who played intramural sports are surveyed then there would be a 1% chance that we would end up falsely concluding that the proportion of all students who played intramural sports who received a degree within six years is smaller than 51%.