In 2011, the national percent of low-income working families had an approximatel

Question

In 2011, the national percent of low-income working families had an approximately normal distribution with a mean of 31.3% (The Working Poor Families Project, 2011). Although it has remained slow, some politicians now claim that the recovery from the Great Recession has been steady and noticeable. As a result, it is believed that the national percent of low-income working families is significantly lower now in 2014 than it was in 2011. To support this belief, a recent 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 a 0.10 significance level, test the claim that the national average percent of low-income working families has improved since 2011.

Question:

Write a few brief sentences to state the type of test that should be performed and state the assumptions and conditions that justify its appropriateness.

Clearly identify and state the null and alternate hypothesis for this test.

Use technology to identify the test statistic and the P-value associated with the hypothesis test; provide these values.

State the decision of the hypothesis test based on a 0.10 significance level.

Provide the appropriate conclusion about the claim that the national average percent of low income working families has improved since 2011.

Explanation / Answer

Hypothesized population proportion p0 = 31.3%

Sample proportion p1 = 29.8%

sample std devn, s1 = 4.1%, alpha = 0.1, n1 = 16

We can do one sample mean t- test instead of one sample proportion z-test. Because for one sample proportion z-test np > 10 which is not satisifed.

what we can do is we can change this problem to one sample mean test instead of proportions, as we know the standard deviation of the sample.

assumptions:

The population should be normally distributed.

the variable that is measured should be ratio or interval based

the sample should be a random sample

all these are satisfied, please note that

H0: mu1 = mu

the sample's national average % of low income is equal to the hypothesized average % of 31.3

H1 mu1 > mu

the national average % of low income is more than the hypothesized average % of 31.3

t-statistic = (mu1-mu)/s/sqrt(n)

= (29.8-31.3)/(4.1/4) = -1.46

t-stat = -1.46

p-value = T.DIST.RT(E2,15) = 0.91

since p-value > 0.05, fail to reject null hypothesis. Hence both the proportions are equal, we don't see any significant improvement in proportion.

ALTERNATIVELY

we can ztest for one sample proportion, if we assume that the population is normally distributed and n=16 is enough for the z-test

In that case

H0: p = p0

H1: p > p0

the sample proportion has improved compared to the hypothesized proportion of 31.3%

z = (p - p0/ sqrt(p0(1-p0)/n) = (29.8%-31.3%)/SQRT(31.3%*68.7%/16)

= -0.1294

p-value = 0.44

since p-value > 0.1, fail to reject null hypothesis

sample proportion 29.8% has not improved, it has remained same as hypothesized national average % as 31.3%

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