Question

In February​ 2008, an organization surveyed 1034 adults aged 18 and older and found that 540 believed they would not have enough money to live comfortably in retirement. Does the sample evidence suggest that a majority of adults in a certain country believe they will not have enough money in​ retirement? Use the

alpha equals 0.05α=0.05 level of significance.

What are the null and alternative hypothesis

What is the p value?

Do not reject or reject?

Answer 1

One-Proportion Z test

The following information is provided: The sample size is N = 1034, the number of favorable cases is X = 540 and the sample proportion is pˉ​=X/N​=540/1034​=0.5222, and the significance level is α=0.05 (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p =0.5 Ha: p >0.5 This corresponds to a Right-tailed test, for which a z-test for one population proportion needs to be used. (2a) Critical Value Based on the information provided, the significance level is α=0.05, therefore the critical value for this Right-tailed test is Zc​=1.6449. This can be found by either using excel or the Z distribution table. (2b) Rejection Region The rejection region for this Right-tailed test is Z>1.6449 (3) Test Statistics The z-statistic is computed as follows: (4) The p-value The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true. In this case, the p-value is p =P(Z>1.4305)=0.0763 (5) The Decision about the null hypothesis (a) Using the traditional method Since it is observed that Z=1.4305 < Zc​=1.6449, it is then concluded that the null hypothesis is Not rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0.0763, and since p=0.0763>0.05, it is concluded that the null hypothesis is Not rejected. (6) Conclusion It is concluded that the null hypothesis Ho is Not rejected. Therefore, there is Not enough evidence to claim that the population proportion p is greater than 0.5, at the 0.05 significance level.

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