Question

For a random sample of 400 students at Aggie University, 10 are vegetarians. test the claim that the proportion of vegetarians at the university differs from 4%. Use 0.10 as the significance level.

Группа выборов ответов

p-value=0.126, evidence not support claim

p-value=0.000, evidence support claim

p-value=0.250, evidence not support claim

p-value=0.025, evidence support claim

p-value=1.000, evidence not support claim

Answer 1

The answer is:

The following information is provided: The sample size is N = 400, the number of favorable cases is X = 10 and the sample proportion is pˉ​=X/N​=10/400​=0.025, and the significance level is α=0.1 (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p =0.04 Ha: p ≠0.04 This corresponds to a Two-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=0.1, therefore the critical value for this Two-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 Two-tailed test is |Z|>1.6449 i.e. Z>1.6449 or 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|>1.5309)=0.1258 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that |Z|=1.5309 < 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=0.1258, and since p=0.1258>0.1, 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 different than 0.04, at the 0.1 significance level.