b3. Sampit D- 8. Ten years ago, the Natural Resources Defense Council conducted
ID: 3325561 • Letter: B
Question
b3. Sampit D- 8. Ten years ago, the Natural Resources Defense Council conducted a study which showed that 1/3 of all bottled water sold in the US does not meet government standards on contaminants (bacteria, pollutants, etc.). Concerned that the situation may be worsening, an agency collected a random sample of 1000 bottles of water of various brands from stores across the country, and found that 360 of them did not meet government standards. At a level of significance a = .05, do the results of the test justify the agency's concern ? (i. e. can the null hypothesis H, : p = 1/3 be rejected in favor of the alternate hypothesis Hy:p> 1/3 with a probability of error a of at most .05 ?) (15 points)Explanation / Answer
Given :-
The sample size is N=1000, the number of favorable cases is X=360, and the sample proportion is p¯=X/N=360/1000=0.36, and the significance level is =0.05
We gives value of p = 1/3 = 0.33
(1) Null and Alternative Hypotheses
The following null and alternative hypotheses need to be tested:
Ho:p=0.33
Ha:p>0.33
This corresponds to a right-tailed test, for which a z-test for one population proportion needs to be used.
(2) Rejection Region
Based on the information provided, the significance level is =0.05, and the critical value for a right-tailed test is zc=1.64.
The rejection region for this right-tailed test is R={z : z>1.64}
(3) Test Statistics
The z-statistic is computed as follows:
z = [ (p¯–po) / (po*(1po)/n) ] = [ (0.36–0.33) / (0.33*(10.33)/1000) ] = 2.018
(4) Decision about the null hypothesis
Since it is observed that z=2.018>zc=1.64, it is then concluded that the null hypothesis is rejected.
Using the P-value approach: The p-value is p=0.0218, and since p=0.0218 < 0.05, it is concluded that the null hypothesis is rejected.
(5) Conclusion
It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion p is greater than po, at the =0.05 significance level.
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