1. Confidence Intervals and Hypothesis Tests a. A sociologist uses a random sample to survey 765 women. She finds that the mean amount of agreement with the statement "criminal offenders should receive harsher sentences" 7 (measured on a scale of 1-10 where 1 is the lowest amount of agreement and 10 is the highest amount of agreement). The sample variance (s') of the variable "criminal offenders should receive harsher sentences" was 5.5. i. Construct a 95% confidence interval for the mean level of agreement with people who break the law should be given stiffer sentences. Interpret this interval ii. Another sociologist says that he knows that in the population from which this sample is drawn, the mean level of agreement with the statement "People who break the law should be given stiffer sentences" is actually 4. He carries out a hypothesis test (alpha level 0.05) to determine whether the population mean is 4. He calculates a z obtained value of 2.50. State his null hypothesis and his critical z. What does he conclude about his null hypothesis? b. A random sample of 180 Queensland residents was asked what the most important priority is for the Queensland government going into 2022. "Stronger sentences for criminal offenders" was the first choice for 75 people. i. Construct a 95% confidence interval for the proportion of people who picked "Stronger sentences for criminal offenders" as their first choice. Interpret this. ii. We tested the null hypothesis that the proportion of people who chose "Stronger sentences for criminal offenders" is 0.59. We calculated a z obtained of -1.82. What did we conclude about the null hypothesis