A large manufacturing company claims that more than 88% of its customers are fully satisfied with the company’s products. Test this claim at the 5% significance level, if a survey found that 360 out of a random sample of 400 of the company’s customers said that they were fully satisfied with the company’s products.
The following null and alternative hypotheses for the population proportion needs to be tested:
"H_0:p\\le0.88"
"H_a:p>0.88"
This corresponds to a right-tailed test, for which a z-test for one population proportion will be used.
Based on the information provided, the significance level is "\\alpha = 0.05," and the critical value for a right-tailed test is "z_c =1.6469."
The rejection region for this right-tailed test is "R = \\{z: z>1.6469\\}."
The z-statistic is computed as follows:
Since it is observed that "z =1.2309<1.6469=z_c," it is then concluded that the null hypothesis is not rejected.
Using the P-value approach:
The p-value is "p=P(Z>1.2309)= 0.10918," and since "p = 0.10918> 0.05=\\alpha," it is concluded that the null hypothesis is not rejected.
Therefore, there is not enough evidence to claim that the population proportion "p" is greater than 0.88, at the "\\alpha = 0.05" significance level.
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