A fiveyearold census recorded that 20 of the families in a l

  A five-year-old census recorded that 20% of the families in a large community lived below the poverty level. To determine if this percentage has changed, a random sample of 400 families is studied and 70 are found to be living below the poverty level. Does this finding indicate that the current percentage of families earning incomes below the poverty level has changed from what it was five years ago? Test with ?=0.05.

(a) Formulate the null and the alternative hypothesis.

(b) State and calculate the test statistics from the data.

(c) Draw a conclusion.

(d) Calculate the P-value.

Solution

(a) Formulate the null and the alternative hypothesis.

Null hypothesis: p=0.2

Alternative hypothesis: p not equal to 0.2

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(b) State and calculate the test statistics from the data.

Z=(phat-p)/sqrt(p*(1-p)/n)

=(70/400-0.2)/sqrt(0.2*0.8/400)

=-1.25

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(c) Draw a conclusion.

Given a=0.05, the critical values are Z(0.025) = -1.96 or 1.96 (from standard normal table)

The rejection regions are if Z<-1.96 or Z>1.96, we reject the null hypothesis.

Since Z=-1.25 is between -1.96 and 1.96, we do not reject the null hypothesis.

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(d) Calculate the P-value.

The p-value= 2*P(Z<-1.25) =0.2113 (from standard normal table)