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Is depression a common problem among college students? Research has shown many college students suffer some form of depression due to the stress of school work, finances, relationships, etc. This is why all schools offer some form of counseling to help those in need. A study claimed that between the years of 2019 and 2020, the percentage of college students suffering from some form of depression was 36%. Suppose you conducted a survey from a random sample of 59 college students and found that 18 suffered from some form of depression. Use a 1% level of significance to determine whether the proportion of college students suffering from some form of depression has changed from 36%.

(a)

What is the level of significance?

State the null and alternate hypotheses. (Enter != for ≠ as needed.)

*H*_{0}:

*p*=0.36

*H*_{1}:

*p*!=0.36

(b)

What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain.

We'll use the Student's *t*, because the sample size is sufficiently large since *np* > 5 and *nq* > 5.We'll use the standard normal, because the sample size is sufficiently large since *np* > 5 and *nq* > 5. We'll use the Student's *t*, because the sample size is not sufficiently large since *np* < 5 and *nq* < 5.We'll use the standard normal, because the sample size is not sufficiently large since *np* < 5 and *nq* < 5.

Use the value of the sample test statistic to find the corresponding *z* value. (Round your answer to two decimal places.)

(c)

Find the *P*-value of the test statistic. (Round your answer to four decimal places.)

*P*-value =

Sketch the sampling distribution and show the area corresponding to the *P*-value.

A graph has a horizontal axis with values from −3.5 to 3.5. The curve enters the window from the left, just above the horizontal axis, goes up and to the right, changes direction over approximately 0 on the horizontal axis, and then goes down and to the right before exiting the window just above the horizontal axis. The area under the curve to the right of −0.9 is shaded.

A graph has a horizontal axis with values from −3.5 to 3.5. The curve enters the window from the left, just above the horizontal axis, goes up and to the right, changes direction over approximately 0 on the horizontal axis, and then goes down and to the right before exiting the window just above the horizontal axis. The area under the curve to the right of 0.9 is shaded.

A graph has a horizontal axis with values from −3.5 to 3.5. The curve enters the window from the left, just above the horizontal axis, goes up and to the right, changes direction over approximately 0 on the horizontal axis, and then goes down and to the right before exiting the window just above the horizontal axis. The area under the curve to the left of −0.9 is shaded.

A graph has a horizontal axis with values from −3.5 to 3.5. The curve enters the window from the left, just above the horizontal axis, goes up and to the right, changes direction over approximately 0 on the horizontal axis, and then goes down and to the right before exiting the window just above the horizontal axis. The area under the curve to the left of −0.9 as well as the area under the curve to the right of 0.9 are both shaded.

(d)

Based on your answers in parts (a) through (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ?????

At the ???? = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.At the ???? = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the ???? = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the ???? = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

(e)

Interpret your conclusion in the context of the application.

There is sufficient evidence, at the 0.01 level, to conclude that the true proportion of college students suffering from some form of depression has changed from 36%.There is insufficient evidence, at the 0.01 level, to conclude that the true proportion of college students suffering from some form of depression has changed from 36%.