Question

Several years ago, 41% of parents who had children in grades K-12 were satisfied with the quality of education the students receive. A recent poll asked 1115 parents who have children in grades K-12 if they were satisfied with the quality of education the students receive. Of the 1115 surveyed, 452 indicated that they were satisfied. Perform an appropriate hypothesis test to assess whether this represents evidence that parents' attitudes toward the quality of education have changed. Use \alpha=0.05α=0.05 level of significance.

1 - Check requirement and state the hypotheses.

2 - Round test statistic to 3 decimals, and p-value to 4 decimals.

3 - State your conclusion in a complete sentence.

Answer 1

One-Proportion Z test

The following information is provided: The sample size is N = 1115, the number of favorable cases is X = 452 and the sample proportion is pˉ​=X/N​=452/1115​=0.4054, 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 =0.41 Ha: p ≠0.41 This corresponds to a Two-tailed test, for which a z-test for one population proportion needs to be used. Conditions:- We have n*p=1115*0.4054=452 and n*p*(1-p) = 663. Both are greater than 5. The samples selected are random. The sample size is large enough (2a) Critical Value Based on the information provided, the significance level is α=0.05, therefore the critical value for this Two-tailed test is Zc​=1.96. 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|>1.96 i.e. Z>1.96 or Z<-1.96 (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|>0.3136)=0.7538 (5) The Decision about the null hypothesis (a) Using the traditional method Since it is observed that |Z|=0.3136 < Zc​=1.96, 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=0.7538, and since p=0.7538>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 different than 0.41, at the 0.05 significance level.