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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
(d) Construct a 95% margin of error for your answer to (c) and provide a brief interpretation.
(c) Use the regression equation to predict a READ score for a student who has a gradepoint average of 3.00.
(b) What proportion of the variance in READ scores is explained by the variance in CGPA?What proportion remains unexplained?
1. Regress READ scores on CGPA.(a) Using the regression output, derive the raw-score regression equation that will allow you to predict a READ score for a given value of CGPA. Provide this equation in the form Y0 ¼ a þ bX.
15. Some studies have found a strong negative correlation between how much parents help their children with homework (X) and student achievement (Y). That is, children who receive more parental help on their homework tend to have lower achievement than kids who receive little or no parental help.
How would you explain this?
*14. It is common to find that the correlation between aviation aptitude test scores (X) and pilot proficiency (Y) is higher among aviation cadets than among experienced pilots.
13. Does a low r necessarily mean that there is little \association" between two variables?(Explain.)
*12. An r of +.60 was obtained between IQ (X) and number correct on a word-recognition test (Y) in a large sample of adults. For each of the following, indicate whether or not r would be affected, and if so, how (treat each modification as independent of the others):(a) Y is changed to number of
11. For a particular set of scores, SX ¼ 3 and SY ¼ 5. What is the largest possible value of the covariance? (Remember that r can be positive or negative.)
10. r ¼ :47, SX ¼ 6, and SY ¼ 4. What is the covariance between X and Y?
9. The covariance between X and Y is 72, SX ¼ 8 and SY ¼ 11. What is the value of r?
*8. Calculate r from Problem 7.
(c) Estimate the new r (before proceeding to Problem 8).
(b) In general, how would this change affect r?
*7. Suppose you change the data in Problem 3a so that the bottom case is X ¼ 1 and Y ¼ 12 rather than X ¼ 1 and Y ¼ 2.(a) Without doing any calculations, state how (and why) this change would affect the numerator of the covariance and, in turn, the covariance itself.
(e) What generalizations do these results permit regarding the effect of linear transformations(e.g., halving each score) on the degree of linear association between two variables?Exercises 141
(d) What is the Pearson r between X and Y? How does this compare with the initial r from Problem 4?
(c) How is the covariance affected by this transformation?
(b) What is the covariance between X and Y?
(a) How do your impressions of the new scatterplot compare with your impressions of the original plot?
6. (a) Using the data in Problem 3, divide each value of X by 2 and construct a scatterplot showing the relationship between X and Y.
*5. What is the covariance for the data in Problem 3?
(a) Interpret r within the context of the coefficient of determination.
*4. (a) Using the data in Problem 3, determine r from both the defining formula and the calculating formula.
(c) Based on visual inspection alone and before proceeding to the next problem, estimate Pearson r from this plot.
(b) Do you detect any outliers or evidence of curvilinearity?
(a) What are your impressions of this scatterplot regarding strength and direction of association?
*3. (a) Prepare a scatterplot for the data below, following the guidelines presented in this chapter.X Y 11 12 9 8 8 10 6 7 4 4 3 6 1 2
2. Why is it important to inspect scatterplots?
1. Give examples, other than those mentioned in this chapter, of pairs of variables you would expect to show:(a) a positive association(b) a negative association(c) no association at all
*16. Consider the effect sizes you computed for Problem 15 of Chapter 5. Interpret these within the context of area under the normal curve, as discussed in Section 6.9.
*14. The mean of a set of z scores is always zero. Does this suggest that half of a set of z scores will always be negative and half always positive? (Explain.)15. X ¼ 20 and S ¼ 5 on a test of mathematics problem solving (scores reflect the number of problems solved correctly). Which represents
*13. The following five scores were all determined from the same raw score distribution (assume a normal distribution with X ¼ 35 and S ¼ 6). Order these scores from best to worst in terms of the underlying level of performance.(a) percentile rank = 84(b) X ¼ 23(c) deviation score = 0(d) T ¼
12. Convert each of the scores in Problem 2 to T scores.
*11. Given a normal distribution with X ¼ 500 and S ¼ 100, find the percentile ranks for scores of:(a) 400(b) 450(c) 380(d) 510(e) 593(f) 678
(d) Between what two scores do the central 80% of scores fall?
(c) What score corresponds to the 40th percentile (P40)?
(b) What score is the 70th percentile (P70)?
(a) What score separates the upper 30% of the cases from the lower 70%?
10. Given a normal distribution of tests scores, with X ¼ 250 and S ¼ 50:
9. In a normal distribution, what is the z score:(a) above which the top 5% of the cases fall?(b) above which the top 1% of the cases fall?(c) below which the bottom 5% of the cases fall?(d) below which the bottom 75% of the cases fall?
*8. In a normal distribution, what z scores:(a) enclose the middle 99% of cases?(b) enclose the middle 95% of cases?(c) enclose the middle 75% of cases?(d) enclose the middle 50% of cases?
*7. In a normal distribution, what proportion of cases fall:(a) outside the limits z ¼ 21:00 and z ¼ 11:00?(b) outside the limits z ¼ 2:50 and z ¼ 1:50?(c) outside the limits z ¼ 21:26 and z ¼ 11:83?(d) outside the limits z ¼ 21:96 and z ¼ 11:96?Exercises 111
6. In a normal distribution, what proportion of cases fall between:(a) z ¼ 1:00 and z ¼ þ1:00?(b) z ¼ 1:50 and z ¼ þ1:50?(c) z ¼ 2:28 and z ¼ 0?(d) z ¼ 0 and z ¼ þ:50?(e) z ¼ þ:75 and z ¼ þ1:25?(f) z ¼ :80 and z ¼ 1:60?
*5. In a normal distribution, what proportion of cases fall (report to four decimal places):(a) above z ¼ þ1:00?(b) below z ¼ 2:00?(c) above z ¼ þ3:00?(d) below z ¼ 0?(e) above z ¼ 1:28?(f) below z ¼ 1:62?
4. Make a careful sketch of the normal curve. For each of the z scores of Problem 3, pinpoint as accurately as you can its location on that distribution.
3. Convert the following z scores back to academic self-concept scores from the distribution of Problem 2 (round answers to the nearest whole number):(a) 0(b) 2.10(c) +1.82(d) .75(e) +.25(f) +3.10
110 Chapter 6 Normal Distributions and Standard Scores reflect a positive academic self-concept). Convert each of the following scores to a z score:(a) 70(b) 90(c) 106(d) 100(e) 62(f) 80
3. Assume XPSATM ¼ 49:2 and SPSATM ¼ 14:3 among juniors nationwide. At what percentile does the average (mean) junior at this school score at nationally?normal curve theoretical versus empirical distribution X 6 1S standard score standardized score derived score z score normal curve table area
2. Use the \Save standardized values as variables"option within the Descriptives procedure to convert the scores on LOCTEST, STEXAM, and PSATM to z scores. Three new variables will automatically be added to your data file(ZLOCTEST, ZSTEXAM, and ZPSATM).Briefly explain how student #40 performed
1. Generate histograms for the variables LOCTEST, STEXAM, and PSATM; check the \Display normal curve" option within the histogram procedure.Briefly comment on how close each distribution comes to normality.
*2. X ¼ 82 and S ¼ 12 for the distribution of scores from an \academic self-concept"instrument that is completed by a large group of elementary-level students (high scores The assessments data file contains student assessment results for three 11th-grade mathematics classes. Among the data are
1. What are the various properties of the normal curve?
(d) What is your impression of the magnitude of the two effect sizes?
(c) Compute the effect size for each of these mean differences.
(b) What is the pooled standard deviation for verbal ability?
(a) What is the pooled standard deviation for mathematics achievement?
*15. Imagine you obtained the following results in an investigation of sex differences among high school students:Mathematics Achievement Verbal Ability Male ðn ¼ 32Þ Female ðn ¼ 34Þ Male ðn ¼ 32Þ Female ðn ¼ 34ÞXM ¼ 48 SM ¼ 9:0 XF ¼ 46 SF ¼ 9:2 XM ¼ 75 SM ¼ 12:9 XF ¼ 78 SF ¼
*14. The mean is 67 for a large group of students in a college physics class; Duane obtains a score of 73.(a) From this information only, how would you describe his performance?(b) Suppose S ¼ 20. Now how would you describe his performance?(c) Suppose S ¼ 2. Now how would you describe his
13. Given: X ¼ 500 and S ¼ 100 for the SAT-CR.(a) What percentage of scores would you expect to fall between 400 and 600?(b) between 300 and 700?(c) between 200 and 800?
12. Determine the sum of squares SS corresponding to each of the following standard deviationsðn ¼ 30Þ:(a) 12(b) 9(c) 6(d) 4.5
(c) What do the results of Problem 11b suggest about the relationship between central tendency and variability?
(b) For each set of scores, compute the mean; compute the variance and standard deviation directly from the deviation scores.
(a) Upon inspection, which show(s) the least variability? the most variability?
*11. Consider the four sets of scores:8; 8; 8; 8; 8 6; 6; 8; 10; 10 4; 6; 8; 10; 12 1004; 1006; 1008; 1010; 1012
9. Why is the variance little used as a descriptive measure?*10. Imagine that each of the following pairs of means and standard deviations was determined from scores on a 50-item test. With only this information, describe the probable shape of each distribution. (Assume a normal distribution unless
Which of the statistics above will be affected by the correction? (Explain.)
*8. After you have computed the mean, median, range, and standard deviation of a set of 40 scores, you discover that the lowest score is in error and should be even lower.
7. If you wanted to decrease variance by adding a point to some (but not all) scores in a distribution, which scores would you modify? What would you do if you wanted to increase variance?
*6. For each of the following statistics, what would be the effect of adding one point to every score in a distribution? What generalization do you make from this? (Do this without calculations.)(a) mode(b) median(c) mean(d) range(e) variance(f) standard deviation
5. Given: S2 ¼ 18 and SS ¼ 900. What is n?
4. Determine the standard deviation for the following set of scores. X: 2.5, 6.9, 3.8, 9.3, 5.1, 8.0.
*3. For each set of scores below, compute the range, variance, and standard deviation.(a) 3, 8, 2, 6, 0, 5(b) 5, 1, 9, 8, 3, 4(c) 6, 4, 10, 6, 7, 3
2. Each of five raw scores is converted to a deviation score. The values for four of the deviation scores are as follows: 4, +2, +3, 6. What is the value of the remaining deviation score?
1. Give three examples, other than those mentioned in this chapter, of an \average" (unaccompanied by a measure of variability) that is either insufficient or downright misleading.For each example, explain why a variability measure is necessary.
(b) How do these two groups compare in terms of variability in MATH scores? (How about central tendency?)
(a) Compute descriptive statistics for MATH(the score on the state-administered mathematics exam), and report the results separately for the group of students who took algebra in the eighth grade and for those who took general math. (You will need to use the \split file" command, which you will
2. Access the sophomores data file.
(b) Does one assessment appear more discriminating than the other? That is, do the two assessments differ in their ability to\spread out" students in terms of their performance?
(a) Generate the mean, minimum, maximum, and standard deviation for QUIZ and ESSAY. Grading for both assessments is based on 100 points.
1. Access the fourth data file, which contains student grades from a fourth-grade social studies class.
16. If the eventual purpose of a study involves statistical inference, which measure of central tendency is preferable (all other things being equal)? (Explain.)
15. From an article in a local newspaper: \The median price for the houses sold was$125,000. Included in the upper half [of houses sold] are the one or two homes that could sell for more than $1 million, which brings up the median price for the entire market." Comment?
*14. Which measure(s) of central tendency would you be unable to determine from the following data? Why?Hours of Study per Night f 5+ 6 4 11 3 15 2 13 1 or fewer 8
13. Where must the mode lie in the distribution of GPAs in Table 2.5?5.68 Chapter 4 Central Tendency
*12. What is the mean, median, and mode for the distribution of scores in Table 2.2?
(b) How would the Mdn and mode be affected by this error?*11. Suppose you were a school psychologist and were interested only in improving the median self-esteem score of children in your school. On which of the following students would you work the hardest: (1) those with the lowest self-esteem
(a) What is the correct value for X?
*10. X ¼ 23, Mdn ¼ 28, mode ¼ 31 for a particular distribution of 25 scores. It was subsequently found that a scoring mistake had been made: one score of 43 should have been a 34.
9. At a local K–6 school, the four K–2 teachers have a mean of 15 students per class, while the five teachers for grades 3–6 have a mean of 18 students per class. What is the mean number of students across the nine teachers in this school?
8. A newspaper editor once claimed that more than half of American families earned a below-average income. Could this claim possibly be correct? (Explain.)
*7. State the likely relative positions of the mean, median, and mode for the following distributions:(a) family income in a large city(b) scores on a very easy exam(c) heights of a large group of 25-year-old males(d) the number of classes skipped during the year for a large group of undergraduate
*6. Comment on the probable shape for each of the following distributions (knowing nothing else about these distributions):(a) X ¼ 52, Mdn ¼ 55, mode ¼ 60(b) X ¼ 79, Mdn ¼ 78, mode ¼ 78(c) X ¼ 50, Mdn ¼ 50, mode ¼ 60; 40(d) X ¼ 28, Mdn ¼ 26, mode ¼ 20
(b) Show that SðX XÞ ¼ 0 for the following sample of scores: 2, 5, 7, 8, 13.
5. (a) What is meant by the \balance point" of a distribution of scores? How is the expression, SðX XÞ ¼ 0, relevant to this concept?
4. Obtain the mean, median, and mode for READ using the \Statistics" option within the Frequencies procedure. Comment on what these scores suggest about the shape of this distribution.
3. Which measure of central tendency do you think best captures the typical CGPA in this sophomore class? Explain.
2. Obtain the actual median and mode for CGPA by way of a frequency distribution.
1. Generate a histogram for CGPA and use it to estimate the mean, median, and mode. Mark your estimates on the graph.
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