Question

Part 3: Hypothesis Testing

In 2011, the national percent of low-income working families had an approximately normal distribution with a mean of 31.3% and a standard deviation of 6.2% (The Working Poor Families Project, 2011). Although it remained slow, some politicians claimed that the recovery from the Great Recession was steady and noticeable. As a result, it was believed that the national percent of low-income working families was significantly lower in 2014 than it was in 2011. To support this belief, a spring 2014 sample of n=16 jurisdictions produced a sample mean of 29.8% for the percent of low-income working families, with a sample standard deviation of 4.1%. Using an α=0.10 significance level, test the claim that the national average percent of low-income working families had improved by 2014.

• Clearly identify the claim and state the null and alternate hypotheses for this test.
• Write a few brief sentences to state the type of test that should be performed based on the hypotheses and the information provided (or not provided) in the narrative above. Additionally, state the assumptions and conditions that justify its appropriateness.
• Use technology to identify the test statistic and the resulting P-value associated with the sample results. Provide these values.
• State, separately, both the decision/result of the hypothesis test, and the appropriate conclusion/statement about the claim.

Reference(s): The Working Poor Families Project. (2011). Indicators and Data. Retrieved from http://www.workingpoorfamilies.org/indicators/

2011 Data
Jurisdiction	Percent of low income working families (<200% poverty level)	Percent of 18-64 year olds with no HS diploma
Alabama	37.3	15.3
Alaska	25.9	8.6
Arizona	38.9	14.8
Arkansas	41.8	14
California	34.3	17.6
Colorado	27.6	10.1
Connecticut	21.1	9.5
Delaware	27.8	11.9
District of Columbia	23.2	10.8
Florida	37.3	13.1
Georgia	36.6	14.9
Hawaii	25.8	7.2
Idaho	38.6	10.7
Illinois	30.4	11.5
Indiana	31.9	12.2
Iowa	28.8	8.1
Kansas	32	9.7
Kentucky	34.1	13.6
Louisiana	36.3	16.1
Maine	30.4	7.1
Maryland	19.5	9.7
Massachusetts	20.1	9.1
Michigan	31.6	10
Minnesota	24.2	7.3
Mississippi	43.6	17
Missouri	32.7	11.1
Montana	36	7
Nebraska	31.1	8.7
Nevada	37.4	16.6
New Hampshire	19.7	7.3
New Jersey	21.2	10.1
New Mexico	43	16.2
New York	30.2	13
North Carolina	36.2	13.6
North Dakota	27.2	5.9
Ohio	31.8	10.3
Oklahoma	37.4	13.2
Oregon	33.9	10.8
Pennsylvania	26	9.4
Rhode Island	26.9	12
South Carolina	38.3	14.2
South Dakota	31	8.7
Tennessee	36.6	12.7
Texas	38.3	17.8
Utah	32.3	9.9
Vermont	26.2	6.6
Virginia	23.3	10.2
Washington	26.4	10.2
West Virginia	36.1	12.9
Wisconsin	28.7	8.5
Wyoming	28.1	8

Answer 1

Let 1 represent the 2014 data and 2 represent the 2011 data

We conduct a lower-tailed t test for independent samples to test if μ1 < μ2

Data:       

n1 = 16      

n2 = 51      

x1-bar = 29.8      

x2-bar = 31.3      

s1 = 4.1      

s2 = 6.2      

Hypotheses:       

Ho: μ1 ≥ μ2       

Ha: μ1 < μ2       

Decision Rule:       

α = 0.1      

Degrees of freedom =   16 + 51 - 2 = 65    

Critical t- score =   -1.294712013     

Reject Ho if t <   -1.294712013     

Test Statistic:       

Pooled SD, s = √[{(n1 - 1) s1^2 + (n2 - 1) s2^2} / (n1 + n2 - 2)] =      √(((16 - 1) * 4.1^2 + (51 - 1) * 6.2^2) / (16 + 51 - 2)) = 5.783464493

SE = s * √{(1 /n1) + (1 /n2)} = 5.78346449271209 * √((1/16) + (1/51)) = 1.657220876     

t = (x1-bar -x2-bar)/SE = (29.8 - 31.3)/1.65722087640197 = -0.9051298     

p- value = 0.184369716      

Decision (in terms of the hypotheses):       

Since -0.9051298 > -1.294712013 we fail to reject Ho   

Conclusion (in terms of the problem):       

There is no sufficient evidence to support the claim that the national average percent of low-income working families had improved by 2014.

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