Question

5. Penn State University wants to determine if students are satisfied with the dorm conditions. A researchers takes random samples of 25 students from each year and the students are asked whether or not they are satisfied with the dorm conditions. The results of the study are given below:

	Freshman	Sophomores	Juniors	Seniors
Satisfied	15	12	9	7
Dissatisfied	10	13	16	18

Is there sufficient evidence of a difference in dorm room satisfaction among the various year levels?

	A.	The data proves that there is a difference in dorm room satisfaction among the various year.
	B.	There is sufficient evidence at the 1% significance level of a difference in dorm room satisfaction among the various years.
	C.	There is sufficient evidence at the 5% significance level, but not at the 1% significance level, of a difference in dorm room satisfaction among the various years.
	D.	With P = 0.1117 there is not sufficient evidence of a difference in in dorm room satisfaction among the various years.
	E.	There is not enough information to answer this question.

6. When analyzing survey results from a two way table, the main distinction between a test of independence and a test for homogeneity is:

	A.	How the degrees of freedom are calculated
	B.	how the expected counts are calculated
	C.	the number of samples obtained
	D.	the number of rows in the two way table
	E.	the number of columns in the two way table.

7. A controversial issue in the sport of professional soccer is the use of instant replay for making difficult goal line devisions. Each person in a representative sample of 102 players, fans, coaches, and officials was asked his or her opinion about the use of instant replay for goal line decisions. The data are summarized in the two way frequency table below.

		Opinion
Category		Favor Use	Oppose Use
	Players	22	2
	Fans	18	6
	Coaches	15	26
	Officials	3	10

In testing to see whether opinion with respect to the use of instant replay is independent of the category of the person interviewed, a chi square test statistic of 27.99 and a p value less than 0.001 were calculated which of the following statements is correct?

	A.	The number of degrees of freedom for the test is 8 - 1 = 7
	B.	The chi square test should not have been used because two of the counts in the table are less than 5.
	C.	The null hypothesis sates that there is an association between category and opinion about the use of instant replay, and the small p value suggests that the null hypothesis should be rejected.
	D.	The small p value suggests that there is evidence of an association between category and the opinion about the use of instant replay.
	E.	The chi square test shows that fans favor the use of instant replay.

8.

A controversial issue in the sport of professional soccer is the use of instant replay for making difficult goal line devisions. Each person in a representative sample of 102 players, fans, coaches, and officials was asked his or her opinion about the use of instant replay for goal line decisions. The data are summarized in the two way frequency table below.

		Opinion
Category		Favor Use	Oppose Use
	Players	22	2
	Fans	18	6
	Coaches	15	26
	Officials	3	10

In testing to see whether opinion with respect to the use of instant replay is independent of the category of the person interviewed, a table of respective frequencies is found. In this table the expected number of professional baseball players opposing the use of instant replay is equal to:

	A.	10.4
	B.	24.1
	C.	11
	D.	6
	E.	8.4

9. A study was performed to examine personal goals of children in grades 4, 5, and 6. A random sample of students was selected in Virginia. The students received a personal questionaire regarding achieveing personal goals. They were asked what they most like to do at school: make good grades, be good at sports, or be popular. The results are presented in the table below by sex of the child.

	Boys	Girls
Make good grades	96	295
Be popular	32	45
Be good at sports	95	40

What type of test is appropriate for this situation?

	A.	Chi square goodness of fit
	B.	Chi square test for independence
	C.	Chi square test for homogeneity
	D.	Two sample t test for difference in means
	E.	Two sample z test for difference in proportions

10.

Recent revenue shortfalls in a midwestern state led to a reduction in the state budget for higher education. To offset the reduction, the largest state university proposed a 25% tuition increase. It was determined that such an increase was needed simply to compenstate for the lost support from the state. Separate random samples of 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors from the university were asked whether they were strongly opposed to the increase, given that it was the minimum increase necessary to maintain the university's budget at current levels. Here are the results.  

Strong Opposed	Freshman	Sophomore	Junior	Senior
Yes	39	36	29	18
No	11	14	21	32

Which hypothesis would be appropriate for performing a chi square test?

	A.	The null hypothesis that is the closer students get to graduation, the less likely they are to be opposed to tuition increases. The alternative is that how close students are to graduation makes no difference in their opinion.
	B.	The null hypothesis is that the mean number of students who are strongly opposed is the same for each of the 4 years. The alternative is the mean is the different for at least 2 of the 4 years.
	C.	The null hypothesis is that the distribution of student opinion about the proposed tuition increase is the same for each of the 4 years at the university. The alternative is that the distribution is different for at least 2 of the 4 years.
	D.	The null hypothesis is that year in school and student opinion about the tuition increase in the sample are independent. The alternative is that these variables are dependent.
	E.	The null hypothesis is that there is an association between year in school and opinion about the tuition increase at the university. The alternative hypothesis is that these variables are not associated.

Answer 1

Solution:-

5) (D) With P = 0.1117 there is not sufficient evidence of a difference in in dorm room satisfaction among the various years.

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis:  There is no difference in in dorm room satisfaction among the various years.

Alternative hypothesis: There is difference in in dorm room satisfaction among the various years.

Formulate an analysis plan. For this analysis, the significance level is 0.01. Using sample data, we will conduct a chi-square test for homogeneity.

Analyze sample data. Applying the chi-square test for homogeneity to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = (r - 1) * (c - 1) = (2 - 1) * (4 - 1)
D.F = 3
E_r,c = (n_r * n_c) / n

Χ² = 5.998

where DF is the degrees of freedom, r is the number of populations, c is the number of levels of the categorical variable, n_r is the number of observations from population r, n_c is the number of observations from level c of the categorical variable, n is the number of observations in the sample, E_r,c is the expected frequency count in population r for level c, and O_r,c is the observed frequency count in population r for level c.The P-value is the probability that a chi-square statistic having 3 degrees of freedom is more extreme than 5.998.

We use the Chi-Square Distribution Calculator to find P(Χ² > 5.998) = 0.112

Interpret results. Since the P-value (0.112) is greater than the significance level (0.01), we fail to reject the null hypothesis.

(D) With P = 0.1117 there is not sufficient evidence of a difference in in dorm room satisfaction among the various years.