Question

In a study of high-achieving high school graduates, the authors of a report surveyed 834 high school graduates who were considered "academic superstars" and 436 graduates who were considered "solid performers." One question on the survey asked the distance from their home to the college they attended.

Assuming it is reasonable to regard these two samples as random samples of academic superstars and solid performers nationwide, use the accompanying data to determine if it is reasonable to conclude that the distribution of responses over the distance from home categories is not the same for academic superstars and solid performers. Use

α = 0.05.

	Distance of College from Home (in miles)
Student Group	Less than 40	40 to 99	100 to 199	200 to 399	400 or More
Academic Superstars	158	157	143	150	226
Solid Performers	105	94	83	65	89

State the null and alternative hypotheses.

H₀: Student group and distance of college from home are independent.
H_a: Student group and distance of college from home are not independent. H₀: Student group and distance of college from home are not independent.
H_a: Student group and distance of college from home are independent.     H₀: The proportions falling into the distance categories are not all the same for the two student groups.
H_a: The proportions falling into the distance categories are the same for the two student groups. H₀: The proportions falling into the distance categories are the same for the two student groups.
H_a: The proportions falling into the distance categories are not all the same for the two student groups.

Calculate the test statistic. (Round your answer to two decimal places.)
χ² =

What is the P-value for the test? (Round your answer to four decimal places.)
P-value =

What can you conclude?

Do not reject H₀. There is not enough evidence to conclude that the proportions falling into the distance categories are not all the same for the two student groups. Reject H₀. There is convincing evidence to conclude that the proportions falling into the distance categories are not all the same for the two student groups.     Reject H₀. There is convincing evidence to conclude that there is an association between student group and distance of college from home. Do not reject H₀. There is not enough evidence to conclude that there is an association between student group and distance of college from home.

Answer 1

Ans:

H₀: The proportions falling into the distance categories are the same for the two student groups.
H_a: The proportions falling into the distance categories are not all the same for the two student groups.

	Observed(O)
Student Group	<40	40-99	100-199	200-399	>400	Total
Academic Superstars	158	157	143	150	226	834
Solid Performers	105	94	83	65	89	436
Total	263	251	226	215	315	1270
	Expected€
Student Group	<40	40-99	100-199	200-399	>400	Total
Academic Superstars	172.71024	164.83	148.413	141.189	206.858	834
Solid Performers	90.289764	86.1701	77.5874	73.811	108.142	436
Total	263	251	226	215	315	1270
	Chi square=(O-E)^2/E
Student Group	<40	40-99	100-199	200-399	>400	Total
Academic Superstars	1.25	0.37	0.20	0.55	1.77	4.14
Solid Performers	2.40	0.71	0.38	1.05	3.39	7.93
Total	3.65	1.08	0.57	1.60	5.16	12.07

Test statistic:

Chi square=12.07

df=(2-1)*(5-1)=4

p-value=CHIDIST(12.07,4)=0.0168

As,p-value<0.05,reject H0.

Reject H₀. There is convincing evidence to conclude that the proportions falling into the distance categories are not all the same for the two student groups.