Question

7. The Supplemental Assistance Nutritional Program is the name for the federal government Food stamp program. At the end of the year 2014 the national statistics were staggering of 114 million households, 23 million were receiving food stamps. Social scientists wish to know if the percentage of California households receiving food stamps is the same as that of Florida. Random samples of 1,000 households are obtained for California and Florida and the number of households receiving food stamps is 180 and 150, respectively.

a. Find the percentage of households in the sample receiving food stamps for California and Florida.

b. Test the hypothesis that the "percentage of households receiving food stamps is the same in California as it is in Florida". Write the appropriate null and alternative hypotheses and use a significance level of 0.05. Make sure to give a decision and write a conclusion.

c. Compute the 95% Confidence Interval for the difference in the percentage of households receiving food stamps in California and Florida.

d. Does the Confidence Interval computed in part c agree with your decision in part b? Answer Yes or No and explain.

Answer 1

a) For California, we have that the sample size is N1​=1000, the number of favorable cases is X1​=180, so then the sample proportion is p^​1​=​X1/N1​​=180/1000​=0.18

For Florida, we have that the sample size is N2​=1000, the number of favorable cases is X2​=150, so then the sample proportion is p^​2​=​X2/N2​​=150/1000​=0.15

The value of the pooled proportion is computed as pˉ​=X1​+X2/N1​+N2​​​= 180+150​/1000+1000=0.165

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:p1​=p2​

Ha:p1​̸​=p2​

This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is \alpha = 0.05α=0.05, and the critical value for a two-tailed test is zc​=1.96.

The rejection region for this two-tailed test is R={z:∣z∣>1.96}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣z∣=1.807≤zc​=1.96, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.0707, and since p=0.0707≥0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion: It is concluded that the null hypothesis Ho is not rejected. Therefore we can conclude that "percentage of households receiving food stamps is the same in California as it is in Florida"​, at the 0.05 significance level.

c) 95% confidence interval is

The 95% confidence interval for −0.003<p1​−p2​<0.063.

d) YES the Confidence Interval computed in part c agree with your decision in part b. Because it does contain zero which is NOT significant. therefore we can conclude that "percentage of households receiving food stamps is the same in California as it is in Florida"