Excel
General instructions for data entry into Excel can be found in Appendix C.
Data Analysis
1) Input a data set. (For practice, we can use one of the data sets in the“Work Problems for the Computer”for this chapter.)
2) Since Excel first needs to know the mean and standard deviation, we will need to find that first. So, select Data Analysis and then Descriptive Statistics. ClickOK.
3) Highlight all of the scores in the distribution and put those quadrant num- bers into theInput Rangebox.
4) Select a location for the output. Use theOutput Rangebox if needed.
5) Make sure to clickSummary Statisticsbefore clickingOK. This should generate a box of descriptive statistics, including the mean and standard deviation of the distribution (the first and fifth statistic generated, respectively).
6) Now select the location for thezscore computation for a given raw score number we wish to transform (typically this is the cell directly adjacent to the right of the raw score), and click thefxkey to the immediate left of the input box at the top of the spreadsheet.
7) Excel uses the term standardize for z scores. Search for and select this term.
8) In theXbox, select the raw score we wish to transform.
9) In theMeanandStandard-devbox, select the appropriate values from the Summary Statistics box we just constructed and click OK. The z score should show up in the selected box.
10) To transform all of the raw scores intozscores, we can use the“autofill” function. However, first highlight the z score already transformed and place a $ in front of the number of the coordinate for both the mean and standard deviation components of the equation (since we do not want those values to change). For instance, if our mean value is found in cell E5, change that to E$5; if our standard deviation value is found in cell E9, change that to E$9. This will keep this value constant for all of the autofill calculations. Then highlight the cell containing the zscore and move the cursor to the lower right of the cell until a “+” appears.
Then drag down to a cell corresponding to the last raw score cell and release. This should produce a column ofzscores corresponding to the adjacent raw scores.
Using Microsoft®Excel and SPSS®to FindzScores 151
SPSS
General instructions for inputting data into SPSS can be found in Appendix C.
Data Analysis
1) Input a data set. (For practice, we can use one of the data sets in the“Work Problems for the Computer”for this chapter.)
2) Once the data has been entered, click Analyze on the tool bar, select Descriptive Statisticsand thenDescriptives.
3) Move the column label containing the data we wish to convert intozscores from the left box to theVariablebox. Also, make sure theSave standar- dized values as variablesbox in the lower left corner is checked. This is vitally important.
4) ClickOK(no other work is needed).
5) The output screen will generate some descriptive statistics. However, once we go back to the data file, we will see a new variable (named identical to the selected variable with a“Z”in front of it) with each raw score transformed into its correspondingzscore.
Key Formulas
Formula for finding the Percentile Rank ofX,PR PR of X= B+ 1 2E
N 100 (Formula 5.1)
Formula for finding X given a Percentile Rank,Xp Xp=L+ N P −F
f h (Formula 5.2)
Formulas for transforming anXScore into azScore
Population Sample
z=X−μ
σ (Formula 5.3a) z=X−M
s (Formula 5.3b) Formulas for transformingzto anXScore
Population Sample
X=μ+zσ (Formula 5.4a) X=M+zs (Formula 5.4b)
Key Terms
Percentile Rank Standard Score
zScore Standard Normal Distribution
Questions and Exercises
1 Using the following frequency distribution, what is the percentile rank of a score of:
a 56 b 60 c 54 d 49
X f
62 3
60 4
58 7
56 12
54 10
49 7
44 6
2 What is azscore, conceptually?
3 Why are somezscores positive values, while others are negative?
4 Think of two examples of variables that are believed to be normally distrib- uted across a population. Defend the answers.
5 Think of two examples of variables that are not believed to be normally dis- tributed across a population. Defend the answers.
6 Transform these scores of a population distribution intozscores.
Raw Scores: 4, 5, 7, 9, 10, 11
7 GivenM =14 ands2= 16, what is thezscore of a raw score of 11?
8 In a distribution whereM= 25 ands= 3, what raw score corresponds to az score of 0.36?
Questions and Exercises 153
Assume normality for all remaining questions.
9 If a distribution has a mean of 130 and a standard deviation of 13, what is the probability of randomly selecting a score above 140?
10 WhenM= 34 ands= 3, what percentage of scores are lower than 27?
11 What is the total percentage of scores that lie beyondzscores of ±1.96?
12 What percentage of scores fall between thezscores ±1.28?
13 What is the z value when the probability of selecting a score at random is:
a At or belowz= 0.4207 b At or belowz= 0.3821 c At or abovez= 0.3192 d At or abovez= 0.0694 e At or belowz= 0.1151 f At or abovez= 0.2946 g At or abovez= 0.4641 h At or belowz= 0.4247 i At or abovez= 0.2119
14 What is the probability of randomly drawing a score between thezscores +0.56 and−1.2?
15 In a distribution withM= 78 ands= 7, what is the probability of selecting a score between 72 and 80?
16 In a distribution having a mean of 123 and a variance of 49, what is the total percentage of scores falling above 130 and below 116?
17 If a standardized anxiety questionnaire has a mean of 25 and a standard deviation of 5, what is the probability that an individual selected at random will score between 20 and 30?
18 A standardized test of reasoning ability has a mean of 70 and a standard devi- ation of 7. The principal of a school would like to identify the best and worst students, as defined by their scores on the test. The best students are those with a percentile rank of 90 and above, and the worst students are those with a percentile rank of 10 and below. What are the raw score cutoffs the prin- cipal should use to identify the two groups of students?
19 Transform the following population of raw scores intozscores.
2, 4, 5, 6, 8, 9
20 For a distribution withM =48 ands= 4, what is the percentile rank of:
a 43 b 57 c 48 d 50 e 47
21 A 100-point final exam is administered in a class whereμ= 78 andσ= 7.
What score did these four students receive?
a Laurie, with a percentile rank of 95%.
b Jennifer, if she is in the 80th percentile.
c Jim, who scored better than 30% of the other students.
d Gus, with a percentile rank of 45%.
22 For a distribution with M= 35 and s = 3, find the percentage of scores that are:
a At or abovez= +1.20 b At or abovez=−0.36 c At or belowz=−0.56 d At or belowz=−0.79 e At or belowz=−1.10 f At or belowz= +0.98 g At or abovez= +0.13
23 Professor Seitz gives a final exam to his abnormal psychology class and finds thatμ= 56 andσ = 5.
a If the passing score is 38, what percentage of students will fail?
b If Professor Seitz wants the“C”category to span the middle 30% of the distribution, what would be the cutoffs?
c What score would serve as the cutoff for an“A”if only the top 10% of the class is to receive an“A?”
24 A student receives a score that corresponds to a percentile rank of 80%.
a What z score corresponds to this rank?
b Given the information available here, can we determine the raw score?
25 A score from a population that is 10 points below the mean corresponds to azscore of−2.50. What is the population standard deviation?
26 A sample score that is 5 points above the mean corresponds to azscore of 2.00. What is the sample standard deviation?
27 For a population with a standard deviation of 15, a raw score of 51 corre- sponds to azof−1.00. What is the population mean?
Questions and Exercises 155
28 For a sample with a standard deviation of 5, a raw score of 31 corresponds to azof 2.00. What is the sample mean?
29 For a population with a mean of 60, a raw score of 61 corresponds to azof 0.20. What is the population standard deviation?
30 For a sample with a mean of 75, a raw score of 60 corresponds to azof
−2.00. What is the sample standard deviation?
31 For a given sample distribution, a raw score of 35 corresponds to a zof
−1.00 and a raw score of 40 corresponds to az of−0.50. Find the mean and standard deviation for this sample.
32 For a given population, a raw score of 72 corresponds to azof 0.20 and a raw score of 84 corresponds to a zof 0.80. Find the mean and standard deviation of the population.
33 For a given sample distribution, a raw score of 16 corresponds to azof– 2.00 and a raw score of 23.5 corresponds to azof 3.00. Find the mean and standard deviation of the sample.
34 For a given population, a raw score of 77 corresponds to azof 2.50 and a raw score of 41 corresponds to azscore of−5.00. Find the mean and stand- ard deviation of the population.
35 Suppose miles traveled per year by American drivers is normally dis- tributed with a mean of 25 000 miles and a standard deviation of 6 000 miles. If we wanted to find the miles traveled that will cut the distri- bution into five equally populated segments, what are the miles trav- eled that define the bottom 20%, the next 20%, and so on up to the top 20%?
36 Suppose the average American household generates 45 lb of garbage per week with a standard deviation of 11 lb. Suppose the local government wants to levy a tax on the worst offenders (top 15%) and offer a tax rebate as incentive for its most conscientious citizens (bottom 28%). What weekly garbage amounts correspond to these cutoffs?
37 For a sample distribution with a mean of 25 and a standard deviation of 4, what raw score corresponds to azscore of−1.75, and what percent of sam- ple scores will be greater than that raw score?
38 For a population of scores with a mean of 99 and a standard deviation of 9, what raw score corresponds to azscore of 1.33, and what percent of scores will be greater than that raw score?
39 For a sample distribution with a mean of 150 and standard deviation of 15, what percent of scores will fall between the values of 170 and 175?
40 For a population of scores with a mean of 1 and a standard deviation of 0.15, what percent of scores will fall between the values of 0.6 and 0.7?
41 Suppose Andrew and Lisa wanted to compare how well they each performed in their respective soccer games. They are, however, on different teams. Since both are defenders and neither score very much, they decided to compare the number of completed passes. Andrew completed 54 passes; his team average was 44 completed passes per player with a standard deviation of 6. Lisa com- pleted 48 passes; her team average was 38 passes with a standard deviation of 7. Which one performed better relative to their teammates?
42 Suppose both Sarah and Justine think the other wastes too much time.
Sarah feels Justine spends too much time on social media compared with others, while Justine feels Sarah spends too much time figuring out what to wear compared with others. Suppose further we know that people average 65 minutes per day on social media (σ=20 minutes) and 15 minutes per day deciding what to wear (σ = 4 minutes). Justine spends 90 minutes a day on social media and Sarah spends 20 minutes deciding what to wear.
Who wastes more time compared with the rest of the population?
43 Using the following grouped frequency distribution, what is the percentile rank of a score of 172?
Class interval Frequency
180–184 7
175–179 11
170–174 16
165–169 15
160–164 11
155–159 9
150–154 7
Questions and Exercises 157
44 For the following grouped frequency distribution, find the percentile rank for anXof 40, 50, 65, and 90.
Class interval Frequency
26–29 17
22–25 13
18–21 23
14–17 27
10–13 25
6–9 12
2–5 9
Computer Work
45 For the following population of scores:
12 15 34 23 32 12 22 21 19 25 14 11 12
11 10 14 15 13 12 16 18 21 29 32 31 30
24 30 29 28 26 21 19 17 16 15 11 10 17
32 30 29 29 28 27 21 14 21 18 16 16 11
20 23 14 15 17 11 21 32 20 20 25 15 17
14 15 23 26 30 24 19 23 22 21 24 17 15
Findμ,σ2,σand convert all raw scores intozscores.
46 For the following sample of scores:
10 5 1 19 13 6 11 12 9 15 17 17 6
4 16 19 8 13 11 7 18 16 7 6 16 2
7 7 11 8 4 11 18 10 14 20 15 4 19
9 3 8 16 5 7 1 19 20 18 12 9 4
9 11 5 15 5 17 17 9 18 1 8 18 6
16 6 12 6 6 18 19 11 18 9 19 17 11
FindM, s2, s and convert all raw scores intozscores.
47 For the following sample of scores:
112 175 344 123 327 412 122 217 419 125 147 411 112 112 108 145 125 183 152 126 188 251 229 382 351 320 243 309 629 283 269 216 193 197 166 153 119 106 173 324 130 279 429 128 277 421 114 217 184 116 167 411 520 223 148 155 127 181 251 322 280 250 225 185 157 164 153 239 266 303 294 196 233 229 621 243 197 156
FindM,s2,sand convert all raw scores intozscores.
48 For the following population of scores:
4.2 8.5 3.4 2.3 3.2 2.2 2.2 2.8 6.9 2.5 1.4 9.1 2.2 5.6 9.1 2.4 4.5 6.3 1.2 3.6 9.8 2.5 2.9 3.2 3.8 3.1 2.4 3.0 2.9 2.8 2.6 2.9 4.9 1.7 4.6 7.5 1.2 7.0 3.7 3.2 3.1 2.9 2.9 2.8 2.7 2.5 2.4 2.3 8.8 3.6 6.6 4.5 2.0 2.3 3.4 5.5 7.7 8.1 2.6 3.2 2.0 2.1 2.5 5.5 6.7 7.4 1.5 2.3 2.6 3.1 2.4 7.9 2.3 2.2 2.9 2.4 4.7 7.5
Findμ,σ2,σ and convert all raw scores intozscores.
Questions and Exercises 159
Part 3
Inferential Statistics Theoretical Basis