STATS ASSIGNMENT FOR MATHEMATICS EXPERT

4. Suppose that in testing the hypothesis Ho: against the alternative H1: , the null hypothesis is rejected at the .05 level. Indicate whether each of the following statements is correct (C) or incorrect (I). (2 points)

_____ There is only a 5% chance that the null hypothesis is true.

_____ The 95% confidence interval for the mean contains the value of 50.

_____ If the population mean was 50, there is only a 5% chance that we would obtain the observed sample mean or anything more extreme.

_____ We are 95% confident that the population mean is not 50.

5. A company specializing in prep courses for college admissions tests (Company A) claims that its clients increase their admissions scores by an average of 50 points as a result of the program. A competing company (Company B) disputes this claim and asks for proof.

Company A takes a random sample of 60 clients, and pre- and post-tests them. The average increase in scores is 48.8 points, with a standard deviation of 9.8 points.

(a) Write an appropriate null and alternate hypothesis for testing the claim of interest. (1 point)

Ho:

H1:

(b) The 95% confidence interval computed by Company A for the average score increase of its clients was (46.3, 51.3). Using this interval, determine whether Company A's claim is supported or not. Explain your conclusion clearly and fully. (1 point)

(c) Give two ways of obtaining a narrower confidence interval. Be explicit. (1 point)

6. Company B (from Q5) also claims that the average score increase of its clients is greater than that of Company A. In support of its claim, it takes a random sample of 60 clients and records the score increase of each. The mean score increase in the sample is 51.6 (s.d. = 10.9). It compares its results to those of the sample of Company A’s clients, given above.

(a) What statistical test is appropriate for Program B’s claim? Put a check next to the correct test. (1 point)

_____ t-test for inference about a population mean

_____ z-test for inference about a population proportion

_____ t-test for inference about the difference in population means using dependent samples

_____ t-test for inference about the difference in population means using independent samples

_____ z-test for inference about the difference in population proportions using independent samples

(b) What effect would each of the following have on the outcome of the hypothesis test for testing Company B’s claim, other things held constant? Indicate whether it would increase the chance of rejection (I), decrease the chance of rejection (D), or would have no effect (NE). (2 points)

_____ Using a larger sample size

_____ Using a higher significance level in testing the hypothesis (e.g., .10 rather than .05)

_____ Constructing a confidence interval rather doing a hypothesis test

_____ Using more heterogeneous samples (i.e., samples with larger variances)

(c) In testing Company B’s claim, the observed value of the test statistic was 1.48, with a p-value of 0.14. What does this p-value mean? Explain clearly and fully. (1 point)

(d) Based on this p-value (not the test statistic), determine whether Company B's claim is supported at the .05 significance level. Explain your answer clearly. (1 point)

(e) Given your conclusion in (d), what type of error (Type I or Type II) could you have made? (1 point)

7. A watchdog group claims that only 30% of workers are willing to act as whistleblowers for fear of retaliation. The CEO of a large company wanted to know if that was true in her company and hired a consultant to investigate. A survey of 250 workers in non-managerial positions was conducted. Workers were presented with a scenario of impropriety, fraud, or mismanagement by their immediate superior and asked if they would report the incident at a higher level. 92 out of the 250 workers said they would report the incident.

(a) Write the appropriate null and alternate hypotheses for testing the claim, assuming a two-tailed test. (1 point)

Ho:

H1:

(b) If the null hypothesis was tested at the .05 level and rejected, what is the probability that the decision to reject was wrong? Explain. (1 point)

(c) The hypothesis in (a) was tested using the sample above, and an observed test statistic of 2.35 was obtained. What is the lowest level of significance at which the null hypothesis could have been rejected (again assuming a two-tailed test)? Show clearly how you obtained your answer. (1 point)

(d) What should the consultant report to the CEO? Write a statement that summarizes the findings from (c). (1 point)

8. The consultant suspected that males and females differ in their willingness to be whistleblowers. In the sample above, 46 of the 140 males and 46 of the 110 females indicated that they would blow the whistle.

(a) Write the null and alternate hypotheses to be tested. (1 point)

Ho:

H1:

(b) The consultant obtained an observed value of the test statistic of 1.46. Draw a conclusion about the hypothesis based on this result. Explain your reasoning and state your conclusion clearly and fully. (2 points)

9. In each of the scenarios below,

write the null and alternate hypotheses to be tested, assuming a two-tailed test;

determine the appropriate statistical procedures for testing the hypothesis (choose from the list below);

(iii) the critical value of the test statistic at the .05 level (show how you obtained that value);

(i) what the researcher’s conclusion would be based on the value of the test statistic given (show clearly how you obtained your answer). Do not simply indicate Reject/Not Reject.

(3 points for each question)

A t-test for inference about a population mean

B z-test for inference about a population proportion

C t-test for inference about the difference in population means using dependent samples

D t-test for inference about the difference in population means using independent samples

E z-test for inference about the difference in population proportions using dependent samples

F z-test for inference about the difference in population proportions using independent samples

(a) To assess the effectiveness of a negative political advertisement against a particular candidate’s opponent, 212 prospective voters were asked whether they intended to vote for the candidate or his opponent. They were then shown the negative advertisement about the candidate’s opponent and asked again whether they would vote for the candidate. Of the 120 who originally supported the candidate, 21 changed their minds after viewing the advertisement and said they would vote for the opponent; of the 92 who originally supported the opponent, 11 changed their minds after viewing the advertisement and said they would vote for the candidate. The observed value of the test statistic for testing the hypothesis of interest was 1.77.

Ho:

H1:

Procedure:

Critical value:

Conclusion:

(b) A researcher interested in studying the personality attributes of last-born children gathered a sample of 20 individuals aged 20-25 who were the youngest in a family of 2-4 children. Each individual completed a scale rating their perception of the degree of conflict amongst siblings they experienced as a child. For each individual in the sample, one of their parents also completed a scale rating their perception of the degree of sibling conflict. The mean perceived conflict score of the last-born individuals was 23.7, while the mean for the parents was 21.5. The means of parents and children were compared to determine if they differed. The observed value of the test statistic was 1.87.

Ho:

H1:

Procedure:

Critical value:

Conclusion:

(c) A researcher studying the effect of background music on concentration skills performed an experiment on 40 college sophomores randomly divided into two groups of 20 each. Both groups were given a reading comprehension test. In one group, pop music at an audible level was played; for the other group, the room was quiet during the test. The mean reading comprehension score under the music conditions was 13.4, while the mean under the quiet condition was 15.9. The variances were not significantly different. An observed test statistic of 2.65 was obtained.

Ho:

H1:

Procedure:

Critical value:

Conclusion:

(d) A researcher interested in the incidence of binge drinking on college campuses and whether there are differences between freshmen and juniors asked 210 freshmen and 180 juniors whether they had engaged in binge drinking in the last month. 43% of the freshmen and 35% of the juniors said they had engaged in binge drinking as defined in the survey. Using these data, a test statistic of 1.61 was obtained.

Ho:

H1:

Procedure:

Critical value:

Conclusion:

10. In a study of the effectiveness of an innovative after-school program for academically at-risk students, a group of 42 students was randomly assigned to one of two programs. Twenty-one students participated in a traditional remedial program, while the other 21 participated in the new after-school program. At the start of the program, all students were administered a scale measuring attitude toward learning. After three months in their programs, all students were re-administered the attitude towards learning scale, along with a standardized achievement test. The data were analyzed using SPSS; the results of the analysis are attached. Analyses are labeled (A), (B), and (C); the questions that follow refer to these labeled analyses.

(A) Comparison of Achievement Scores:

Group Statistics

	Type of Program	N	Mean	Std. Deviation	Std. Error Mean
Ach	Traditional Remedial	21	36.90	3.897	.851
	New After-School	21	35.14	3.439	.751

Independent Samples Test

	Levene's Test for Equality of Variances	t-test for Equality of Means
	F	Sig.	t	df	Sig. (2-tailed)	Mean Difference	Std. Error Difference	95% Confidence Interval of the Difference
								Lower	Upper
Ach	Equal variances assumed	.182	.672	(1)	(2)	.128	(3)	1.134	-.531	4.054
	Equal variances not assumed			1.553	39.390	.128	1.762	1.134	-.532	4.056

(B) Comparison of pre- and post-test Attitude Scores: New After-School Program

Paired Samples Statistics

	Mean	N	Std. Deviation	Std. Error Mean
Pair 1	Atttitude Toward Learning posttest	24.3810	21	2.45919	.53664
	Atttitude Toward Learning pretest	23.2857	21	2.07709	.45326

Paired Samples Test

	Paired Differences	t	df	Sig. (2-tailed)
	Mean	Std. Deviation	Std. Error Mean	95% Confidence Interval of the Difference
				Lower	Upper
Pair 1	Atttitude Toward Learning posttest - Atttitude Toward Learning pretest	1.095	2.119	(4)	.131	2.060	2.368	(5)	.028

(C) Comparison of pre- and post-test Attitude Scores: Traditional program

Paired Samples Statistics

	Mean	N	Std. Deviation	Std. Error Mean
Pair 1	Atttitude Toward Learning posttest	24.0952	21	2.09535	.45724
	Atttitude Toward Learning pretest	23.3810	21	2.37647	.51859

Paired Samples Test

	Paired Differences	t	df	Sig. (2-tailed)
	Mean	Std. Deviation	Std. Error Mean	95% Confidence Interval of the Difference
				Lower	Upper
Pair 1	Atttitude Toward Learning posttest - Atttitude Toward Learning pretest	.714	2.194	.479	-.284	1.713	1.492		.151

(a) Fill in the three values missing in Analysis (A) and the two values missing in Analysis (B). These are indicated (1) through (5) in the tables. (3 points)

(b) State the null and alternate hypotheses being tested in Analysis (A). (1 point)

Ho:

H1:

(c) What does the F statistic in Analysis (A) refer to and what does the “Sig.” value next to it tell you? What implications does this have for interpreting the table? (1 point)

(d) What conclusion would you draw about the effectiveness of the new program based on the results of Analysis (A)? Explain clearly and fully, indicating which values you used to draw your conclusion and how you interpreted them. (1 point)

(c) State the null and alternate hypotheses being tested in Analysis (B). (1 point)

Ho:

H1:

(d) What conclusions would you draw based on the results of Analyses (B) and (C)? Indicate which values you used to draw you conclusions and how you interpreted them. (2 points)

11. In a study to determine the impact of sleep deprivation on students’ academic performance, seniors in a suburban high school were surveyed and asked how many hours nightly they sleep on average. Students who reported they slept less than five hours a night were classified as sleep-deprived; students who reported sleeping seven hours or more were classified as normal sleepers. Of this group, 36 were classified as sleep-deprived and 51 as normal sleepers. The scores of these students on the statewide achievement test were recorded. The means of the two groups are given below.

Normal sleepers: Mean = 23.2, SD = 4.5, N = 51

Sleep-deprived: Mean = 20.6, SD = 4.9, N = 36

(a) Test an appropriate hypothesis using a two-tailed procedure with a significance level of .05. Carry out ALL steps in the hypothesis testing procedure and clearly state your conclusions. (4 points)

(b) Do your results indicate that sleep deprivation causes lower mean achievement? Explain your answer. (1 point)

(c) Construct a 90% confidence interval for the mean difference in the achievement scores of the two groups (2 points). Interpret this interval (say what the numbers mean) and state what conclusions you would draw based on the interval. (1 point)

12. A college using a new method of teaching calculus wishes to evaluate the effectiveness of the method. A class of 24 first-year college students was randomly divided into two sections, one of which was taught using the new method and the other taught using the old method. Scores on the end-of-semester exam for the two groups of students are given below.

New	91	80	85	87	74	76	83	89	87	75	79	88
Old	86	78	80	83	72	75	82	76	84	72	74	87

(a) Write down the null and alternate hypotheses of interest. (1 point).

(b) Enter the data into SPSS and carry out an appropriate hypothesis test. Report the table of means and standard deviations and the table of test results. (2 points)

(c) What conclusion would you draw on the basis of the test of equality of variances? Be specific about how you used the results in the output to draw you conclusion. (1 point)

(d) Interpret the “sig.value” of the test-statistic for testing the hypothesis in (a), i.e., say exactly what this value means. (1 point)

(e) What would you conclude about the relative effectiveness of the two methods? Indicate clearly what evidence you used to reach this conclusion. (1 point)

(f) Repeat the analysis assuming that the students were matched on prior mathematics achievement. Report the table of results (2 points), interpret the sig. value (1 point), and state your conclusions clearly and completely (1 point)

(g) Interpret the confidence interval provided in the paired samples test output. Be specific about what information the interval provides. (1 point)

(h) Based on the results of the paired samples t-test, do the data support the claim that the new method yields a mean increase in achievement of 7 points? Explain your answer. (1 point)

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