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nonparametric statistical inference
Fundamentals Of Statistical Reasoning In Education 3rd Edition Theodore Coladarci, Casey D. Cobb, Edward W. Minium, Robert C. Clarke - Solutions
4. You select a random sample of 10 observations and compute s, the estimate of s. Even though there are 10 observations, s is really based on only nine independent pieces of information.(Explain.)
*3. A random sample of five observations is selected. The deviation scores for the first four observations are 5, 3, 1, and 2.(a) What is the fifth deviation score?(b) Compute SS and sX for the sample of all five observations.
2. When would S (Formula 5.2) and s (Formula 13.1) be very similar? very different?(Explain.)
*1. Ben knows that the standard deviation of a particular population of scores equals 16.However, he does not know the value of the population mean and wishes to test the hypothesis H0: m ¼ 100. He selects a random sample, computes X, s, and sX, and proceeds with a t test. Comment?
(b) Determine whether mean per-pupil expenditure is significantly different from $5500.
(a) Determine whether mean teacher salary is significantly different from $32,000.
2. Access the fiscal data file, which contains average teacher salaries and per-pupil expenditures from 60 school districts in a southeastern state. Use the one-sample t test to conduct the following tasks.
1. Access gosset, a data set used by W. S. Gosset circa 1908. The file contains one variable, ADDHRS, which represents the additional hours of sleep gained by 10 patients after exposure to laevohysocyamine hydrobromide. Using the one-sample t test, determine whether the experimental treatment
10. For a random sample, X ¼ 83 and n ¼ 625; assume s ¼ 15.(a) Test H0: m ¼ 80 against H1: 6¼ 80 ða ¼ :05Þ. What does this tell you about m?(b) Construct the 95% confidence interval for m. What does this tell you about m?(c) Which approach gives you more information about m? (Explain.)
*9. (a) If a hypothesized value of m falls outside a 99% confidence interval, will it also fall outside the 95% confidence interval for the same sample results?(b) If a hypothesized value of m falls outside a 95% confidence interval, will it also fall outside the 99% confidence interval for the
(c) Explain your answers to Problems 8a and 8b.
(b) What would the outcome be for a test of H0: m ¼ 60?
(a) Suppose for the same set of sample results H0: m ¼ 48 were tested using a ¼ :01(two-tailed). What would the outcome be?estimation point estimate interval estimate confidence level confidence interval confidence limits 95% confidence interval 99% confidence interval Exercises 253
*8. The 99% confidence interval for m is computed from a random sample. It runs from 43.7 to 51.2.
7. The interval width is much wider in Problem 6a than in Problem 6d.What is the principal reason for this discrepancy? Explain by referring to the calculations that Formula (12.1) entails.
6. Construct a confidence interval for m that corresponds to each scenario in Problems 15a and 15c–15e in Chapter 11.
5. Consider Problem 4 in Chapter 11, where X ¼ 48, n ¼ 36, and s ¼ 10.(a) Construct a 95% confidence interval for m.(b) Construct a 99% confidence interval for m.
4. Repeat Problems 2a and 2b with n ¼ 9 and then with n ¼ 100. What generalization is illustrated by a comparison of the two sets of answers (i.e., n ¼ 9 versus n ¼ 100)?
*3. Explain in precise terms the meaning of the interval you calculated in Problem 2b.Exactly what does \95% confidence" refer to?
(d) What generalization is illustrated by a comparison of your answers to Problems 2b and 2c?
(c) Construct the 99% confidence interval for her population mean score.
(b) Construct the 95% confidence interval for her population mean score.
*2. The results for Rachel’s sample in Problem 1 is X ¼ 33:10 ðn ¼ 36Þ.(a) Calculate sX.
*1. The national norm for third graders on a standardized test of reading achievement is a mean score of 27 ðs ¼ 4Þ. Rachel determines the mean score on this test for a random sample of third graders from her school district.(a) Phrase a question about her population mean that could be answered
How many of your intervals captured m?
would capture the actual population mean?
How many confidence intervals did you expect
3. Use the information above to construct ten 68%confidence intervals. Record them in the table below.
2. Select a random sample of 25 cases from the population of 521 students. (Use the Select Cases procedure, which is located within the Data menu.) Calculate the mean and standard error for READ. Repeat this entire process nine times and record your results.
1. Compute the mean READ score for the entire population of 521 students. Record it in the top row of the table below.
20. Suppose a researcher wishes to test H0: m ¼ 100 against H1: m > 100 using the .05 level of significance; however, if she obtains a sample mean far enough below 100 to suggest that H0 is unreasonable, she will switch her alternative hypothesis to H1: 6¼ 100 (a = .05)with the same sample data.
*19. Josh wants to be almost certain that he does not commit a Type I error, so he plans to set a at .00001.What advice would you give Josh?
18. What is the relationship between the level of significance and the probability of a Type I error?
What is the probability that she has committed a Type I error? (Explain.)
17. On the basis of her statistical analysis, a researcher retains the hypothesis, H0 : m ¼ 250.
*16. A researcher plans to test H0: m ¼ 3:50. His alternative hypothesis is H1: 6¼ 3:50. Complete the following sentences:(a) A Type I error is possible only if the population mean is ———.(b) A Type II error is possible only if the population mean is ———.
*15. Given: m ¼ 60, s ¼ 12. For each of the following scenarios, report za, the sample z ratio, its p value, and the corresponding statistical decision. (Note: For a one-tailed test, assume that the sample result is consistent with the form of H1.)(a) X ¼ 53, n ¼ 25, a ¼ :05 (two-tailed)(b) X
14. Under what conditions is a directional H1 appropriate? (Provide several examples.)
13. To which hypothesis, H0 or H1, do we restrict the use of the terms retain and reject?
12. Can you make a direct test of, say, H0 6¼ 75? (Explain.)
*11. Explain in general terms the roles of H0 and H1 in hypothesis testing.
*10. Repeat Problems 9a–9c, but for H1: 6¼ 500. 3(d) Compare these results with those of Problem 9; explain why the two sets of results are different.(e) What does this suggest about which is more likely to give significant results: a twotailed test or a one-tailed test (provided the direction
9. State the critical values for testing H0: m ¼ 500 against H1: m < 500, where(a) a ¼ :01(b) a ¼ :05(c) a ¼ :10
8. Mrs. Grant wishes to compare the performance of sixth-grade students in her district with the national norm of 100 on a widely used aptitude test. The results for a random sample of her sixth graders lead her to retain H0: m ¼ 100 (a ¼ :01) for her population.She concludes, \My research proves
*7. Consider the generalization from Problem 6. What does this generalization mean for the distinction between a statistically significant result and an important result?
*6. Compare the results from Problem 5 with those of Problem 4. What generalization does this comparison illustrate regarding the role of n in significance testing? (Explain.)
*5. Repeat Problems 4a–4e, but with n ¼ 100.
*4. Let’s say the personnel director in Problem 1 obtained X ¼ 48 based on a sample of size 36. Further suppose that s ¼ 10, a ¼ :05, and a two-tailed test is conducted.(a) Calculate sX.(b) Calculate z.(c) What is the probability associated with this test statistic?(d) What statistical
*3. Suppose that the personnel director in Problem 1 wants to know whether the keyboarding speed of secretaries at her company is different from the national mean of 50.(a) State H0.(b) Which form of H1 is appropriate in this instance—directional or nondirectional?(Explain.)(c) State H1.(d)
2. The personnel director in Problem 1 finds her sample results to be highly inconsistent with the hypothesis that m ¼ 50 words per minute. Does this indicate that something is wrong with her sample and that she should draw another? (Explain.)
*1. The personnel director of a large corporation determines the keyboarding speeds, on certain standard materials, of a random sample of secretaries from her company. She wishes to test the hypothesis that the mean for her population is equal to 50 words per minute, the national norm for
4. Test the hypothesis that seniors who reported spending fewer than three hours of homework per week score below average on READ.
3. Test the hypothesis that seniors who reported spending at least 5 1/2 hours on homework per week score higher than the national average on READ, MATH, and SCIENCE. READ, MATH, and SCIENCE represent standardized test scores measured in T-score units (m ¼ 50, s ¼ 10).
2. Test the hypothesis that the GPAs among seniors are, on average, different from those of juniors.Assume that for juniors, m ¼ 2:70 and s ¼ :75.
1. Use SPSS to generate the mean for the variable GPA. GPA represents the grade-point averages of courses taken in math, English language arts, science, and social studies.
(c) If you randomly select a sample (n ¼ 25), what is the probability that the sample median will fall above 55 (assume a normal sampling distribution)?(d) For a normal population where m is unknown, which is likely to be a better estimate of m: the sample mean or the sample median? (Explain.)
(b) It is known from statistical theory that if the population distribution is normal, then sMdn ¼1:253s ffiffiffi n pWhat does sMdn stand for (give the name)? In conceptual terms, what is sMdn?
(a) Describe the operations you would go through to obtain a fairly accurate picture of the sampling distribution of medians for samples of size 25. (Assume you have unlimited time and resources.)
16. A population of personality test scores is normal with m ¼ 50 and s ¼ 10.
(b) In this regard, would 100 be reasonable as a possible value of m?
(a) Would 107 be reasonable as a possible value for m in light of the sample mean of 108? (Explain in terms of probabilities.)
*15. You randomly select a sample (n ¼ 50) from the population in Problem 14 and obtain a sample mean of X ¼ 108. Remember: Although you know that s ¼ 15, you don’t know the value of m.
*14. Suppose for a normally distributed population of observations you know that s ¼ 15, but you don’t know the value of m. You plan to select a random sample (n ¼ 50) and use the sample mean to estimate the population mean.(a) Calculate sX.(b) What is the probability that the sample mean will
13. Suppose you don’t know anything about the shape of the population distribution of ratings used in Problems 11 and 12. Would this lack of knowledge have any implications for solving Problem 11? Problem 12? (Explain.)
(c) Can you see an advantage of using large samples in attempts to estimate the population mean from the mean of a random sample? (Explain.)
(b) What is the effect of this larger sample on the limits within which the central 95% of sample means fall?
(a) What is the effect of this larger sample on the standard error of the mean?
12. Repeat Problem 11h using a sample of size 100.
(h) What are the centrally placed limits such that the probability is .95 that the sample mean will fall within those limits?
(g) What sample mean has such a low value that the probability is .05 of obtaining one as low or lower?
What is the probability of obtaining a sample mean:(b) above 6.6?(c) as extreme as 3.8?(d) below 4.4?(e) between the population mean and .5 points below the mean?(f) no more than .5 points away from the population mean (in either direction)?
11. A population of peer ratings of physical attractiveness is approximately normal with m ¼ 5:2 and s ¼ 1:6. A random sample of four ratings is selected from this population.(a) Calculate sX.
(c) What would be the shape of the distribution of sample means? (How do you know?)
(b) What would be the standard deviation of the sample means?
(a) What would be the mean of the resulting sample means?
*10. Suppose you collected an unlimited number of random samples of size 36 from the population in Problem 9.
(f) Within what limits would the central 95% of all possible sample means fall?
(e) What sample mean has such a high value that the probability is .01 of obtaining one as high or higher?
(d) What is the probability that the sample mean will differ from the population mean by 4 points or more (in either direction)?
(c) What is the probability that the sample mean will fall more than 8 points above the population mean of 100?
(b) What is the probability that the sample mean will fall above 92?Exercises 211
(a) Calculate the standard error of the mean.
*9. Given: m ¼ 100 and s ¼ 30 for a normally distributed population of observations. Suppose you randomly selected from this population a sample of size 36.
*8. What are the key questions to be answered in any statistical inference problem?
7. What are the three defining characteristics of any sampling distribution of means?
6. Explain on an intuitive basis why the sampling distribution of means for n ¼ 2 selected from the \flat" distribution of Figure 10.4a has more cases in the middle than at the extremes.(Hint: Compare the number of ways an extremely high or an extremely low mean could be obtained with the number
*5. Suppose you did not know Formula (10.2) for sX. If you had unlimited time and resources, how would you go about obtaining an empirical estimate of sX for samples of three cases each drawn from the population of Problem 4?
(b) What would you expect the sampling distribution of means to look like for samples of size 150 selected from this population? State the principle used to arrive at your answer.
(a) Suppose you want to obtain a fairly accurate picture of the sampling distribution of means for random samples of size 3 drawn from this population. Suppose also that you have unlimited time and resources. Describe how, through repeated sampling, you could arrive at such a picture.
4. A certain population of observations is bimodal (see Figure 3.10b).
*3. A researcher conducting a study on attitudes toward \homeschooling" has her assistant select a random sample of 10 members from a large suburban church. The sample selected comprises nine women and one man. Upon seeing the uneven distribution of sexes in the sample, the assistant complains,
2. After considering the sampling problems associated with Problem 1, your friend decides to interview people who literally are \on the street." That is, he stands on a downtown sidewalk and takes as his population passersby who come near enough that he might buttonhole them for an interview. List
(c) Do you believe this sample is representative of the apparent population? If not, in what ways might this sample be biased?
(b) What is the sample in this instance?
(a) What population does the newscaster appear to have in mind?
*1. \The average person on the street is not happy," or so claimed the newscaster after interviewing patrons of a local sports bar regarding severe sanctions that had been imposed on the state university for NCAA infractions.
3. Open a new (empty) data file in SPSS. Input the 20 sample means in a column, naming the variable S_MEANS. Compute its mean and standard deviation (i.e., the mean and standard deviation of the sample means). Also generate a histogram for S_MEANS and compare it to the histogram of the population
2. Select a random sample of 25 cases from the population of 521 students. To do so, use the Select Cases procedure, which is located within the Data menu. Calculate the mean for CGPA. Repeat this entire process 19 times and record your results.
1. Compute the mean CGPA score for the entire population of 521 students; generate a histogram for CGPA.
a) the first student falls at least 100 SAT-CR points away from the mean (in either direction)?(b) both students obtain SAT-CR scores above 700?(c) the first student obtains a score above 650 and the second student obtains a score below 450?(d) both students fall above 650 or both students fall
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