Psych 124 1Jan. 5, 2005

Practice Problems (1/5/05)

Next Wednesday (1/12/05) will have a short (20-30 min) quiz on the following topics: (a) frequencydistributions, (b) histograms, (c) terminology from chapter 1, (d) chi-square goodness-of-fit test, (e) chi-square testfor independence, and (f) extracting probabilities, such as, P χ3

2 > 3.67( ) , from the Table of the chi-squaredistribution.

To help you understand the above material, I offer many problems for you to attempt. You should be ableto do most of the problems by the end of Week 1; a few problems pertain to topics to be covered in Week 2. Thesolutions to many, not all, of the problems are given. If you want to check a solution, feel free to ask me, the TA’s,or your friend in the class. I hope you try some of the problems for which the solution is not given - this helps youdevelop confidence in what you do know. The more problems you work, the more you will learn about thisimportant subject and the more confident you will be in performing statistical tests. Feel free to form ‘studygroups’, in which you discuss the concepts and formulae, and then see who can work the most problems.

Suggested practice problems from the text (with answers):• Aron & Aron, Chapter 1, Set 1: 1, 4, 6, 8, 10• Study Guide, Chapter 1: 24• Aron & Aron, Chapter 14, Set 1: 3a, 3c, 4a, 4c, 7a, 7c• Study Guide, Chapter 14: 22, 23a, 23c

Constructing univariate and bivariate frequency distributions

A small sample of residents in a county provided data onthree variables, Region (home in ‘suburban’ or ‘urban’ region ofcounty), Attitude (‘for’, ‘neutral’, or ‘against’ Proposition A), andOccupation (‘professional’ or ‘non-professional’). We use thefollowing numerical coding of the values of each variable:‘suburban’ = 0, ‘urban’ = 1; ‘for’ = 1, ‘neutral’ = 2, ‘against’ = 3;‘non-professional’ = 0, ‘professional’ = 1. The data are givenbelow. (Note: In preparing your data for analysis, e.g., bycomputer, it is very common to arrange the data such that each rowcorresponds to a single subject; and each column corresponds toone of the variables being studied.)

Exercises

1.(a) Construct the (univariate) frequency distribution of Region.(b) Repeat for Attitude. (Ans. f1 = 5, f2 = 5, f3 = 6.) (c) Repeat forOccupation.

2. (a) Construct the (bivariate) frequency distribution of Region and Occup. (b) Repeat for Region and Attitude. (c)Repeat for Occupation and Attitude.

3. Use the appropriate bivariate distribution to answer thefollowing. (a) How many people have homes in ‘suburban’ andare ‘neutral’? (Ans. 2.) (b) How many people are ‘professional’and are ‘for’? (Ans. 3.) (c) How many people are from ‘urban’regions or are ‘for’? (Ans. 14.)

A note on joint and marginal frequencies. The distribution in 2(b) above has 2 rows and 3 columns and,therefore, it has 2x3 = 6 cells. The numbers in each cell are called joint frequencies – they count the joint events,Region = i and Attitude = j, i = 1, 2, and j = 1, 2 or 3. In addition to the 6 cells, this table (like every other table) hastwo margins, one in which the row totals (or row marginal frequencies) are put, and the other in which the columntotals are put. Each marginal frequency has a meaning. For example, in 2(b), the ‘6’ in the Row 1 margin states that

DataResID Region Attit Occup1 0 1 02 0 1 13 0 2 14 0 1 15 0 2 16 0 1 17 1 1 08 1 3 19 1 3 010 1 2 011 1 3 012 1 2 013 1 3 014 1 3 115 1 2 016 1 3 0

Ans. for 2(b):Attitude

Region 1 2 3 Total0 4 2 0 61 1 3 6 10Total 5 5 6 16

Psych 124 2Jan. 5, 2005

there are 6 persons in the sample who have Region = 0 (i.e., ‘suburban’ homes); the ‘5’ in the Column 2 marginstates that there are 5 persons in the sample with a ‘neutral’ attitude.

Distribution of Chi-Square for given probability levels and degrees of freedom (df)

Probabilitydf .98 .95 .90 .80 .70 .50 .30 .20 .10 .05 .021 .001 .004 .016 .064 .15 .46 1.07 1.64 2.71 3.84 5.412 .04 .10 .21 .45 .71 1.39 2.41 3.22 4.61 5.99 7.823 .19 .35 .58 1.01 1.42 2.37 3.67 4.64 6.25 7.82 9.844 .43 .71 1.06 1.65 2.20 3.36 4.88 5.99 7.78 9.49 11.675 .75 1.15 1.61 2.34 3.00 4.35 6.06 7.29 9.24 11.07 13.39

Practice Quiz Set #1

1. Michelle is interested in the distance between a student’s home and the university he or she attends. She createsthree zones (that is, categories of distance): less than 100 miles, 100 to 1000 miles, and more than 1000 miles. Hernull hypothesis is that universities draw students with equal frequency from each of these three zones. To examinethis hypothesis, she visits the Registrar’s Office at a certain university, collects the relevant data and constructs thefrequency distribution for distance. The number of students coming from each of the zones are 1050, 1010, and 940,respectively. (a) Test Michelle’s null hypothesis, using a significance level of .05. (Please state your argumentsclearly). (60) (b) Would you say that the observed distances are generally greater or are generally lower than thosehypothesised by the instructor? Why (only 1 sentence, please)? (No statistical test is necessary for this part.) (20)

(Ans. (a) χ2 = 2.5+0.1+3.5 = 6.2, with k-1 = 2 df. Critical value at the 5% sig. level is 5.99. Since 6.2 > 5.99, rejectH0. (b) More students than were hypothesised come from ‘near’ (1050 vs. 1000), and fewer students than werehypothesised come from ‘far’ (940 vs. 1000). Therefore, the observed distances are generally lower than thosehypothesised by the instructor.)

2. Jackie thinks that the distance between a student’s hometown and the university that he or she attends might berelated to the ‘prestige’ of the university. She finds a Low Prestige university and a High Prestige university. Ineach school she counts how many students live within 100 miles of campus, between 100 and 1000 miles, and morethan 1000 miles from campus. The data were asfollows:

(a) Compute the marginal frequencies. (10) (b)Describe verbally what each of the two univariate(marginal) frequency distributions tells us. (20) (c) Test the null hypothesis that there is no relationship between‘distance’ and ‘prestige’. (Please use the .05 level of significance, and state your arguments clearly.) (80) (d) Givea good description of what the cell frequencies tell us. (20)

(Ans. (b) The row marginal frequencies suggest that the High prestige school is about twice as big as the Lowprestige school. The column marginal freqs. suggest that, overall, students are evenly spread over ‘near’,‘moderately far’ and ‘far’ distances. (c) E11 = 336.7, E12 = 333.3, etc.; χ2 = 6.6, which is ‘large’. So, reject H0;conclude there is a relationship between Prestige and Distance. (d) At the Low prestige school, the mode is ‘< 100’,i.e., a plurality of students are from ‘near’. At the High prestige school, the plurality is from ‘far’. In simpler terms,Low prestige schools tend to draw students from close by, whereas High prestige schools have a broader appeal anddraw students from far and wide.)

3. χk2 means “chi-square with k degrees of freedom”. Also Pr ob χk

2 < c( ) + Prob χ k2 > c( ) = 1 , for any number c. (a)

What is Pr ob χ12 > .46( ) ? (10) (b) What is Pr ob 1< χ3

2 > 6.25( ) ? (15) (c) Find c such that Pr ob χ22 < c( ) = 0.05 ?

(15) (Ans. (a) 0.498. (b) 0.8 - 0.1 = 0.7. (c) c = 0.103.)

DistancePrestige <100 100-1000 >1000Low 360 340 310High 640 650 700

Psych 124 3Jan. 5, 2005

4. In each of two colleges, students were classified according to the region where they were born, i.e., East, South,West, South-West, and Middle. The distrns across Region for each College are as shown. Compare the twodistrns. (10)

(Ans. In Coll. 1 the mode is ‘West’(Stanford?), and in Coll 2 the mode is‘South’ (Vanderbilt?). In Coll 2, the rf atthe mode is 0.65 (> 0.50 in Coll 1), andthere are 0 students from the ‘West’. Thissuggests that Coll 1 draws from a wider geog. area.)

Practice Quiz Set 2

1. Lara, who was writing a term paper on ‘Attitudes towards the life at Stanford’, wanted some information quicklyabout the attitudes of each Class (Class = Fresher, Sophomore, Junior or Senior). Attitude was measured on a 10-point scale with ‘0’ = ‘very negative’, and ‘10’ = ‘very positive’. She surveyed a random sample of 4 sophomores, 4juniors and 4 seniors. The data were: Sophomores - 9, 1, 8, 6. Juniors - 2, 2, 3, 5. Seniors - 8, 5, 5, 6.

(a) Please write a sentence or two for Lara’s paper on Classes and Attitude. (No test of a null hypothesis is requiredhere.) (40) (Ans. Use mean and range as descriptive stats. Sophomores and Seniors have mildly positive attitudesto SU ( x = 6), and Juniors have a negative attitude ( x = 3). Sophomores are most variable in attitudes - range = 9 -1 = 8 (compared to the range of 3 for the other 2 groups.)

2. Dave has many resources. He tackles the issue of ‘Attitudes and Class’ by computing the frequency distributionof the Class of students who attend different types of events. First, he surveys the students at a few ‘fun, non-academic’ events and found that 32 of these students were Freshers, 24 were Sophomores, 40 were Juniors, and 24were Seniors. Second, he surveys the students at ‘serious, academic’ events and found that 8 were Freshers, 8 wereSophomores, 8 were Juniors, and 56 were Seniors.

(a) Compute the relative frequency distribution of Class for ‘fun’ events, and the relative frequency distribution ofClass for ‘serious’ events. (20)

(b) Compare the two distributions that were just computed in part (a). (20) (Ans. Compare the mode and the rf atmode of the 2 distrns. Most of the students at Serious events are seniors. At Fun events, there is a more evendistribution across classes.)

(c) Please offer a brief, cogent explanation for Dave’s data. (20)

3. Elizabeth has a null hypothesis that, at ‘fun, non-academic’ events, there are equal numbers of students from eachClass. She uses Dave’s data (as given in Qu. 2 above) to test her null hypothesis.

(a) Please carry out this test of her hypothesis at the .05 level of significance, stating your arguments andconclusions carefully. (50) (Ans. Ei = 120/4 = 30, etc. χ2 = 5.9, less than the critical value of 7.82. So retain H0.)

4. χa2 means ‘chi-square with a degrees of freedom’. Also Pr ob χa

2 < c( ) + Prob χ a2 > c( ) = 1 , for any number c. (a)

What is Pr ob χ12 > 1.07( )? (15) (Ans. 0.7.) (b) What is Pr ob 1.01< χ 3

2 > 6.25( )? (20) (Ans. 0.7.) (c) Find c suchthat Pr ob χ4

2 < c( ) = .20 . (20) (Ans. 1.649.)

5. Experience has shown that the number of hours that a student spends studying for a quiz has the chi-squaredistribution with 5 degrees of freedom. In a group of 200 students, about how many would you expect to study forthree hours or less for a quiz? (25) (Ans. Pr ob χ5

2 < 3( ) = 1− 0.7 = 0.3 . So E = 200x0.3 = 60.)

RegionCollege East South West S-West MiddleColl 1 .15 .05 .50 .10 .20Coll 2 .20 .65 .00 .05 .10

Psych 124 4Jan. 5, 2005

Practice Quiz Set 3

A. An initiative, labelled Measure I, has been put on the ballot for the upcoming elections in a State. Two groups ofadults are sampled and asked their opinion of Measure I; one group of adults resides in the State (‘Resident’), andthe other group does not reside in the State (‘Non-resident’). The opinion of each adult is recorded as ‘For’,‘Undecided’, or ‘Against’. Among the ‘Resident’ sample, 60 are ‘For’ the Measure, 10 are ‘Undecided’, and 30 are‘Against’ it. Among the ‘Non-resident’ sample, 15 are ‘For’ the Measure, 25 are ‘Undecided’, and 10 are ‘Against’it.

1. Compute the relative frequency distribution of opinion about Measure I for (a) ‘Residents’, and (b) ‘Non-residents’. (30)

2. Please write a short, cogent comparison of the distributions found in (1a) and (1b) above. (No statistical test isrequired.) (10) (Ans. Residents tend to be For the Measure, with a substantial minority against. Non-residents tendto be undecided.)

B. A student of politics has a theory that, among ‘Non-residents’ (defined in Qu. A above), 60% are ‘Undecided’about Measure I, and the remaining adults are equally divided between being ‘For’ and being ‘Against’ it (i.e., 20%are ‘For’ and 20% are ‘Against’ it).

3. Use the ‘Non-resident’ data, given in Qu. A above, to test the student’s theory. (Note that, in this case, the theoryis the null hypothesis.) (Please use the .05 level of significance, and state your arguments clearly.) (50) (Ans. χ2 =3.33, which is less than 5.99. So, retain H0.)

C. Another student has the interesting hunch that, among ‘Residents’, opinion about Measure I might depend on theadult’s length of residence in the State. Accordingly, the student randomly samples 100 ‘Residents’ and asks themfor (a) their length of residence in the State, coded as ‘< 5 years’, ‘5-10 years’, or ‘> 10 years’; and (b) their opinionabout Measure I, coded as ‘For’ or ‘Against’. (The ‘Undecided’ category was not used in this study.) In thesample, 60 are ‘For’, and 40 are ‘Against’ Measure I. Of the 60 who are ‘For’, 30 have resided in the State ‘< 5years’, and 25 have resided ‘5-10 years’ (and the remainder have resided ‘> 10 years’). Of the 40 who are ‘Against’,20 have resided in the State ‘< 5 years’, and 15 have resided ‘5-10 years’ (and the remainder have resided ‘> 10years’).

4. Construct the contingency table from these data, and calculate the marginal frequencies. (20)

5. Test the null hypothesis that there is no relationship between length of residence and opinion about Measure I.(Please use the .05 level of significance, and state your arguments clearly.) (50) (Ans. χ2 = 0.5; retain H0.)

6. Offer a brief description of (a) the marginal frequencies, and (b) the observed relationship between length ofresidence and opinion. (20) (Ans. (a) Row marginal freqs show that the majority are For; the Col marginal freqsshow that the plurality have resided in State for < 5 yrs. (b) No relationship between lenght of resid and opinion.)

D. χa2 means ‘chi-square with a degrees of freedom’. Also Pr ob χa

2 < c( ) + Prob χ a2 > c( ) = 1 , for any number c.

(7) What is Pr ob χ12 > 1.07( )? (Ans. 0.3.) (8) What is Pr ob 1.39 < χ2

2 > 4.61( ) ? (Ans. 0.4) (9) Find c such thatPr ob χ4

2 > c( ) = .20 . (60) (Ans. 5.99.)