Question

A researcher would like to determine if the proportion of households without health insurance coverage differs with household income. Suppose the following data were collected from 700700 randomly selected households. Complete parts a through c.	Heath Insurance
	Household Income	Yes	No
	Less than​ $25,000	55	22
	​$25,000 to​ $49,999	145	40
	​$50,000 to​ $74,999	201	40
	​$75,000 or more	177	20

a. Using alphaαequals=​0.01, perform a​ chi-square test to determine if the proportion of households without health insurance differs by income bracket.

Choose the correct null and alternative hypotheses below.

A.

Upper H 0H0​:

Not all​ p's are equal

Upper H 1H1​:

p 1 equals p 2 equals p 3 equals p 4p1=p2=p3=p4

B.

Upper H 0H0​:

p 1 equals p 2 equals p 3 equals p 4p1=p2=p3=p4

Upper H 1H1​:

p 1 not equals p 2 not equals p 3 not equals p 4p1≠p2≠p3≠p4

C.

Upper H 0H0​:

p 1 not equals p 2 not equals p 3 not equals p 4p1≠p2≠p3≠p4

Upper H 1H1​:

p 1 equals p 2 equals p 3 equals p 4p1=p2=p3=p4

D.

Upper H 0H0​:

p 1 equals p 2 equals p 3 equals p 4p1=p2=p3=p4

Upper H 1H1​:

Not all​ p's are equal

What is the test​ statistic?

chi squaredχ2equals=nothing

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

What is the critical​ value?

chi Subscript 0.01 Superscript 2χ20.01equals=nothing

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

State the conclusion.

▼

Do not reject

Reject

Upper H 0H0.

There is

▼

sufficient

insufficient

evidence that the proportion of households without health insurance differs by income bracket.

b. Interpret the meaning of the​ p-value.

What is the​ p-value?

​p-valueequals=nothing

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

What does the​ p-value mean? Select the correct choice and fill in the answer box to complete your choice.

​(Round to one decimal place as​ needed.)

A.There is a

nothing​%

chance of observing a test statistic value greater than the actual test statistic value if there is no difference in the proportion of households without health insurance.

B.There is a

nothing​%

chance of rejecting the null hypothesis when it should not be rejected.

C.Given a very large number of​ samples, there is a

nothing​%

chance of observing a sample with the given data.

c. How does income appear to impact the likelihood that a household has insurance​ coverage?

A.

The proportion of households without health insurance is always uniform and does not depend on income bracket.

B.The proportion of households without health insurance

does not differdoes not differ

by income bracket.

C.

The proportion of households without health insurance increases as income increases.

D.The proportion of households without health insurance

differs by income bracket.

Click to select your answer(s).

Answer 1

(a) The correct null and alternative hypotheses:

D. H0: p1=p2=p3=p4

H1: Not all​ p's are equal

>> The test​ statistic: χ2 = 16.27

>> The critical​ value: χ2​​​0.01 = 11.34

>> Reject H0. There is sufficient evidence that the proportion of households without health insurance differs by income.

(b) p-value = 0.001

>> A.There is a 1% chance of observing a test statistic value greater than the actual test statistic value if there is no difference in the proportion of households without health insurance.

(c) D. The proportion of households without health insurance differs by income.

** Software Outputs:

Actual Values:
55 22
145 40
201 40
177 20

Expected Values:
63.58 13.42
152.757 32.2429
198.997 42.0029
162.666 34.3343

Chi-Squared Values:
1.15786 5.48557
0.393915 1.86625
0.0201583 0.0955039
1.26315 5.98445

Chi-Square = 16.2669

Degrees of Freedom = 3

p-value = 0.000999707