The effort to reward city students for passing Advanced Placement tests is part

Question

The effort to reward city students for passing Advanced Placement tests is part of a growing trend nationally and internationally. Financial incentives are offered in order to lift attendance and achievement rates. One such program in Dallas, Texas, offers $100 for every Advanced Placement test on which a student scores a three or higher (Reuters, September 20, 2010). A wealthy entrepreneur decides to experiment with the same idea of rewarding students to enhance performance, but in Chicago. He offers monetary incentives to students at an inner-city high school. Due to this incentive, 122 students take the Advancement Placement tests. Twelve tests are scored at 5, the highest possible score. There are 49 tests with scores of 3 and 4, and 61 tests with failing scores of 1 and 2. Historically, about 100 of these tests are taken at this school each year, where 8% score 5, 38% score 3 and 4, and the remaining are failing scores of 1 and 2.

1) Conduct a hypothesis test that determines, at the 5% significance level, whether the monetary incentive has resulted in a higher proportion of scores of 5, the highest possible score.Be sure you include all of the following required steps of the hypothesis testing procedure (Show work).

a) clearly state the null and alternative hypothesis. b) Develop a decision rule based on a selected level of significance. c)Compute the requiered test statistic using sample data. d) clearly state a decision: reject or do not reject. e) State a conclusion that expresses the results of the test.

2)  Define a Type I Error for this problem. What is the probability of making a Type I Error?

3) Define a Type II Error for this problem.

Explanation / Answer

1) Conduct a hypothesis test that determines, at the 5% significance level, whether the monetary incentive has resulted in a higher proportion of scores of 5, the highest possible score. Be sure you include all of the following required steps of the hypothesis testing procedure (Show work).

Here, we have to use z test for two population proportions.

a) Clearly state the null and alternative hypothesis.

The null and alternative hypothesis for this test is given as below:

Null hypothesis: H0: The percentage of score 5 is same for program when financial incentives are available and financial incentives are not available.

Alternative hypothesis: Ha: The percentage of score 5 when financial incentives are available is higher than the percentage of score 5 when incentives are not available.

H0: p1 = p2 versus Ha: p1 > p2

b) Develop a decision rule based on a selected level of significance.

We are given a level of significance = alpha = 0.05

So, critical z value = 1.6449

Reject H0 when test statistic Z > 1.6449

Or

Reject H0 when p-value < 0.05

c) Compute the required test statistic using sample data.

The test statistic formula is given as below:

Z = (P1 – P2)/sqrt[(P1Q1/N1)+(P2Q2/N2)]

We are given

P1 = 0.12

P2 = 0.08

Q1 = 1 – 0.12 = 0.88

Q2 = 1 – 0.08 = 0.92

N1 = 100

N2 = 100

Z = (0.12 – 0.08)/sqrt((0.12*0.88/100)+(0.08*0.92/100))

Z = 0.944911

P-value = 0.1724

d) Clearly state a decision: reject or do not reject.

For the given test,

Test statistic Z = 0.9449 < Critical value Z = 1.6449

P-value = 0.1724 > alpha value = 0.05

So, we do not reject the null hypothesis

e) State a conclusion that expresses the results of the test.

There is insufficient evidence to conclude that the percentage of score 5 when financial incentives are available is higher than the percentage of score 5 when incentives are not available.

2) Define a Type I Error for this problem. What is the probability of making a Type I Error?

Type I error is defined as the probability of rejecting null hypothesis that the percentage of score 5 is same for program when financial incentives are available and financial incentives are not available; even though it is true.

P(Type I) = 0.1724

3) Define a Type II Error for this problem.

Type II error is defined as the probability of do not rejecting null hypothesis that the percentage of score 5 is same for program when financial incentives are available and financial incentives are not available; even though it is not true.

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