Question

A survey was conducted two years ago asking college students their top motivations for using a credit card. The percentages are shown in the table to the right. Also shown in the table is the observed frequency for these motivations from a current random sample of college students who use a credit card. Complete parts a through c below.	Response	Old Survey​ %	New Survey ​ Frequency, f
	Rewards	2727​%	112112
	Low rates	2323​%	9797
	Cash back	2222​%	109109
	Discounts	77​%	4646
	Other	2121​%	6161

a. Using

alpha equals 0.05α=0.05​,

perform a​ chi-square test to determine if the probability distribution for the motivations for using a credit card changed between the two surveys.What is the null​ hypothesis,

Upper H 0H0​?

A.The distribution of motivations is

112112

​rewards,

9797

low​ rates,

109109

cash​ back,

4646

​discounts, and

6161

other.

B.

The distribution of motivations differs from the claimed or expected distribution.

C.The distribution of motivations is

2727​%

​rewards,

2323​%

low​ rates,

2222​%

cash​ back,

7 %7%

​discounts, and

2121​%

other.

D.

The distribution of motivations follows the normal distribution.

What is the alternate​ hypothesis,

Upper H 1H1​?

A.

The distribution of motivations does not follow the normal distribution.

B.

The distribution of motivations is​ 20% rewards,​ 20% low​ rates, 20% cash​ back, 20%​ discounts, and​ 20% other.

C.

The distribution of motivations is the same as the claimed or expected distribution.

D.

The distribution of motivations differs from the claimed or expected distribution.

Calculate the test statistic.

chi squaredχ2equals=20.4620.46

​(Round to two decimal places as​ needed.)Determine the critical​ value,

chi Subscript alpha Superscript 2χ2α​,

and the rejection region.

chi Subscript alpha Superscript 2χ2αequals=9.4889.488

​(Round to three decimal places as​ needed.)

Choose the correct rejection region below.

A.

chi less than or equals chi Subscript alpha Superscript 2χ≤χ2α

B.

chi squared greater than or equals chi Subscript alpha Superscript 2χ2≥χ2α

C.

chi squared less than chi Subscript alpha Superscript 2χ2<χ2α

D.

chi squared greater than chi Subscript alpha Superscript 2χ2>χ2α

b. Determine the​ p-value and interpret its meaning.

​p-valueequals=. 003.003

​(Round to three decimal places as​ needed.)

Interpret the​ p-value.

The​ p-value is the probability of observing a test statistic

greater than

less than

equal to

greater than

the test​ statistic, assuming

the distribution of the variable is the same as the given distribution.

at least one expected frequency differs from 5.

the distribution of the variable differs from the given distribution.

the distribution of the variable differs from the normal distribution.

the expected frequencies are all equal to 5.

the distribution of the variable is the normal distribution.

the distribution of the variable is the same as the given distribution.

c. What conclusion can be drawn about the motivation behind the use of a credit card by college students between the two​ surveys?

Reject

Do not reject

Reject

Upper H 0H0.

At the

55​%

significance​ level, there

is

is not

is

enough evidence to conclude that the distribution of motivations

differs from

is the same as

differs from

the claimed or expected distribution.

Click to select your answer(s).

Answer 1

for null hypothesis : option C is correct

H1: option D is correct

2)

applying chi square goodness of fit test:

	relative	observed	Expected	residual	Chi square
category	frequency(p)	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
1	0.27	112.0	114.75	-0.26	0.066
2	0.23	97.0	97.75	-0.08	0.006
3	0.22	109.0	93.50	1.60	2.570
4	0.07	46.0	29.75	2.98	8.876
5	0.21	61.0	89.25	-2.99	8.942
total	1.000	425	425		20.4591
test statistic X2 =		20.46

critical​ value χ2α = 9.488

p value =

0.000

The​ p-value is the probability of observing a test statistic greater than the test​ statistic, assuming the distribution of the variable is the same as the given distribution.

c)

Reject H0 At the 5% significance​ level, there is enough evidence to conclude that the distribution of motivations differs from the claimed or expected distribution.