Question

stats

Q1. Hypothesis testing using a Z test (14 points)

A professor has been teaching introductory statistics for many years and the final exam performance (30 points total) has been very consistent from class to class and the scores have been normally distributed. Overall, the whole data base (i.e. population) of final exam scores has a mean (μ) of 20 points and a standard deviation (σ) of 5 points. Because 20 out of 30 is only about 67%, the professor would like to try a new format to see if student performance could be improved.

So in the most recent class, students were allowed to make two attempts (instead of one attempt) at the final exam. There were 64 students and the class average on the final exam was 21.5. The professor would like to run a hypothesis test to see if this sample of students (with two attempts) performed significantly better than the past population (with one attempt). In other words, the hypothesis was a comparison between the population of one attempt and the population of two attempts (represented by the sample of 64 students).

The professor is predicting an increase of final exam score with two attempts, so the hypotheses should be directional, and the test should be one-tailed. α = .05.

e. Calculate the Z statistic (which indicates where our sample mean is located on the sampling distribution) (2 points total: 1 if the process is correct but the result was calculated incorrectly) f. Determine the critical value for Z. Explain how you come up with the answer. (1 point: .5 for the answer and .5 for the rationale) g. Compare the Z statistic with the critical Z value (that is, “is the Z statistic more extreme than the critical Z?”), and then draw a conclusion about the result of the hypothesis test (do you “reject” or “fail to reject” the null hypothesis?) (2 points: 1 for Z statistic comparison; 1 for hypothesis test decision) h. Write 1-2 sentences to answer the research question (you can use the wording from the hypotheses or explain it in another way) (1 point) i. Calculate the standardized effect size for this test. (2 points total: 1 if the process is correct but the result was calculated incorrectly)

Answer 1

Solution:-

e)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: μ < 20

Alternative hypothesis: μ > 20

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the sample mean is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is aone-sample z-test.

Analyze sample data. Using sample data, we compute the standard error (SE), z statistic test statistic (z).

SE = s / sqrt(n)

S.E = 0.625

z = (x - μ) / SE

z = 2.40

f) z_critical = 1.645

g) The Z statistic more extreme than the critical Z.

where s is the standard deviation of the sample, x is the sample mean, μ is the hypothesized population mean, and n is the sample size.

The observed sample mean produced a z statistic test statistic of 2.40.

Thus the P-value in this analysis is 0.008.

Interpret results. Since the P-value (0.008) is less than the significance level (0.05), we have to reject the null hypothesis.

h) Reject H0, From the above test we have sufficient evidence in the favor of the claim that sample of students (with two attempts) performed significantly better than the past population.

i) The standardized effect size for this test is 0.30.

$d=\frac{x-\mu }{S.D}$

$d=\frac{21.5-20 }{5}$

d = 0.30