Local Municipalities in the United Kingdom

Question 1 

It is known that, nationally, prosecutors working for local municipalities average 13.5 years of experience in their specialties, with a standard deviation of 7.6 years. The district attorney of a western state is interested in determining whether or not its prosecutors have less experience than the national average. A random sample of 150 prosecutors from this state shows a mean of only 10.9 years of experience. 

Solution 

a. State the research and null hypotheses to test whether or not prosecutors in this state have less experience than the national average.

Let µ be the average experience of a prosecutor, and  be the sample mean.

Research : The prosecutors in this state have less experience than the national average; 

Null : The prosecutors have an average of 13.5 years of working experience: 

b. Using an alpha level of .01, test this hypothesis.

Assumptions : The sample is large enough (n ≥ 30) and that the random sample comes from a population that is normally distributed.

Type of hypothesis test : The hypothesis is one tailed

Sampling distribution/test statistic : Z statistic

Test Statistic : 

Interpretation/Conclusion : Since the value of the test statistic (-4.1899) is less than  we reject the null hypothesis with 99% confidence and conclude that there is sufficient evidence to support the claim that prosecutors have less experience than the national average.

Question 2 

A colleague of yours offers the following research hypothesis: There is a lower proportion of college graduates among first-generation Asian Americans than second-generation Asian Americans. She then presents you with this table 

First generation	Second generation	Obtained Z Statistic
P1=0.50	P2=0.55	-2.50
N1=2684	N2=566

Solution 

a. What is type of hypothesis test is this (one-tailed or two-tailed)? 

The hypothesis is one tailed.

b. Based on the obtained Z statistic, what can you conclude about the difference in the proportion of college graduates between the two groups?

At 99% level of significance, we would reject the null hypothesis since the value of test statistic is less than  . The conclusion would be that there is a statistically significant difference in proportions of college graduates between the two groups

Question 3

For each of the following situations, determine whether a one- or a two-tailed test is appropriate and state the research and null hypotheses.

Solution 

a. You are interested in finding out if the average household income of residents in your state is different from the national average household. According to the U.S. Census, for 2011, the national average household income is $50,054. 

Type of test : two tailed test 

Research : The average household income is not equal to $50,054. 

 , where µ is the average household income of the population 

Null : The average household income is equal to $50,054 

 . 

b. You believe that students at small liberal arts colleges attend more parties per month than students nationwide. It is known that nationally undergraduate students attend an average of 3.2 parties per month. The average number of parties per month will be calculated from a random sample of students from small liberal arts colleges. 

Solution 

Type of test : one tailed (upper) 

Research : students at small liberal arts colleges attend more than 3.2 parties per month. 

 , where µ is the average number of parties an undergraduate student attends per month. 

Null : The average number of parties an undergraduate student attends per month are 3.2 

Question 4 

One way to check on how representative a survey is of the population from which it was drawn is to compare various characteristics of the sample with the population characteristics. A typical variable used for this purpose is age. The 2010 GSS of the American adult population found a mean age 49.28 years and a standard deviation of 17.21 for its sample of 4,857 adults. Assume that we know from Census data that the mean age of all American adults is 37.2 years. 

Solution 

a. State the research and null hypotheses for a two-tailed test. 

Research : The mean age of American adults is not equal to 37.2 years,  , where µ is the mean age of American adults 

Null : The mean age of American adults is 37.2 years,  

b. Using an alpha level of .01, test this hypothesis. 

Assumptions : The random sample is drawn from a population that is normally distributed 

Type of hypothesis test : two tailed 

Sampling distribution/test statistic (Z or t): Z statistic 

Test statistic :  

Interpretation/conclusion : We reject the null hypothesis with 99% confidence since the value of the test statistic (48.9182) is greater than and we conclude that there is sufficient evident at α=0.01 to support the claim that the average age of American adults is not 37.2.