Soc 274, Section 02: Social Statistics

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Soc 274, Section 02: Social Statistics

Jeremiah Coldsmith

Summer 09, Final Exam

Name: ______

For each question show your work, so I can give you partial credit. Also make sure you answer all the parts of the question. For all calculations, label and circle your answers!

1) The following are the years of education completed for 15 employees at a local manufacturing plant.

10, 12, 12, 14, 12, 16, 16, 15, 14, 12, 13, 18, 20, 14, 12

a. Generate a Frequency Table, including frequency, cumulative frequency, percentage, and cumulative percentage.

b. Draw a histogram or bar chart to represent the frequency distribution of these employees’ years of education completed.

c. Calculate the Mode, Median, and Mean (If appropriate. If it is not appropriate, explain why it isn’t.)

Mode = 12

Median: Middle score = 16/2 = 8 Value of the 8th score is 14. Median = 14

Mean = (10 + 12 + 12 + 12 + 12 + 12 + 13 + 14 + 14 +14 + 15 + 16 + 16 + 18 + 20)/15 = 210/15 = 14

d. Calculate the Range (If appropriate. If it is not appropriate, explain why it isn’t.)

20 – 10 = 10 years of education

e. Calculate the Deviations from the Mean and the Sum of the Squared Deviations. (Don’t forget to label and circle the answers. Make sure I know you know which numbers are which of these.)

f. Calculate the Variance and the Standard Deviation.

V = 98/15 = 6.53 OR v = 98/14 = 7

SD = √6.53 = 2.56 OR s.d. = √7 = 2.64

2) What is the probability of each of the following?

a. Rolling a 3 on an 8 sided die.

p(3) = 1/8 = .125

b. Rolling a 3 on a 12 sided die.

p(3) = 1/12 = .083

c. Rolling an even number on a 6 sided die.

p(2 OR 4 OR 6) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2 = .5

d. Rolling two 9s on two 10 sided dice at the same time.

p(9 AND 9) = 1/10 * 1/10 = 1/100 = .01

e. Rolling an odd number on two consecutive rolls of an 8 sided die.

p((2 OR 4 OR 6 OR 8) * (2 OR 4 OR 6 OR 8)) = (1/8 + 1/8 + 1/8 + 1/8) * (1/8 + 1/8 + 1/8 + 1/8) =

4/8 * 4/8 = 1/2 * 1/2 = 1/4 = .25

Also = 16/64 = 1/4 = .25

3) A test designed to evaluate reading comprehension ability is normally distributed with a mean of 10 and a standard deviation of 1.5. (Note: The scores are truly continuous. Round the scores to the nearest hundredths.)

a. what percent of test takers scored lower than someone who scored an 11.5?

Z = (11.5 – 10)/1.5 = 1.5/1.5 = 1

50 + 34.13 = 84.13% scored below someone who scored 11.5.

b. what percent of test takers scored higher than someone who scored an 8?

Z = (8 – 10)/1.5 = –2/1.5 = –1.33

50 + 40.82 = 90.82% scored higher than someone who scored 8.

c. what percent of test takers scored between the person who scored 11.5 and the one who scored 8?

34.13 + 40.82 = 74.95% of people scored between 8 and 11.5.

d. what would someone need to score in order to have 25% of test takers score better than them?

Z = .67

X = (.67 * 1.5) + 10 = 1.005 + 10 = 11.005

Someone would need to score about an 11 to have 25% of people score better than them.

e. what are the scores that bound the middle 95% of test takers?

Z = ± 1.96

X = (1.96 * 1.5) + 10 = 2.94 + 10 = 12.94

X = (–1.96 * 1.5) + 10 = –2.94 + 10 = 7.06

7.06 and 12.94 bound the middle 95% of scores.

4) A team of political sociologists are concerned that the racial makeup of their most recent sample of union members is not representative of the population of union members. They believe that the union member population is 75% white, 20% black, and 5% other. Their sample of 670 registered voters has 565 whites, 85 blacks, and 20 others. The pollsters want to be 99% sure the sample is different from their expectations.

Calculate the appropriate statistic and let the pollsters know if their sample matches their expectations or not.

HR: The sample does not match their expectations.

H0:The sample matches their expectations.

Level of Significance: .01

Degrees of Freedom: k – 1 = 3 – 1 = 2

Critical Value: 9.2103

Calculate the χ2: (Note: Label and Circle the χ2 value.)

What decision do you make about the Hypotheses?

Because the calculated χ2 is further out in the tail than the critical value, we reject the null hypothesis that the sample matches the researchers’ expectations.

What conclusion can you draw from that decision?

Therefore, at the .01 level of significance the sample does not match the expectations of the researchers.

5) Clinical psychologists are working on a new anger management therapy. They believe this new therapy will decrease aggressive behaviors of those with anger problems. They begin by observing how often their subjects become angered when confronted with a frustrating situation. Then, they perform the new therapy on the subjects. Finally, they observe the subjects again in situations similar to those prior to the therapy to see how often the subjects become angered now. The psychologists want to be 99% sure that the therapy decreases the number of aggressive behaviors exhibited by their subjects. Do the appropriate statistical test and interpret the results.

s.d.: √(11.3334/5) = √2.2667 = 1.5056

Level of Significance: .01 One-Tailed, Negative

d.f.: 6 – 1 = 5

Critical Value: –3.365

HR: The therapy decreases the number of aggressive acts.

H0:The therapy increases or keeps the number of aggressive acts the same.

Work for the statistical test:

t = –1.6667/(1.5056/√6) = –1.6667/(1.5056/2.4495) = –1.6667/.6146 = –2.7118

Decision concerning the hypotheses:

Because the calculated t is closer to the mean than the critical value, we fail to reject the null hypothesis that therapy increases or keeps the number of aggressive acts the same.

Conclusion based on the decision:

Therefore, at the .01 level of significance, the decrease in aggressive acts after the therapy is not statistically significant.

6) A student who has had a sociology statistics course is trying to decide which of two on-line merchants to buy a product from. The prices, shipping costs, and time to deliver are exactly the same. The final thing the student would like to check is how well the two merchants are rated by previous customers. He would like to be 95% sure there is a difference in the ratings of the two stores before deciding from which one to buy. Do the appropriate statistical test and tell the student if the two stores are rated differently.

v1: 3.82/4 = .955

v2: .3/3 = .1

HR: There is a difference in the two stores’ ratings.

H0:There is not a difference in the two stores’ ratings.

d.f.1: 4

d.f.2: 3

d.f.T: 7

Level of Significance: .05 Two-Tailed

Critical Value:±2.365

m1: 8

m2: 9

v1: .955

v2: .1

vpooled: (4/7) .955 + (3/7) .1 = .5714 (.955) + (.4286) .1 = .5457 + .0429 = .5886

vm1:.5886/5 = .1177

vm2: .5886/4 = .1472

vdifference:.1177 + .1472 = .2649

sdifference:√.2649 = .5147

Work for the Statistical Test:

t = (8 – 9)/.5147 = –1/.5147 = –1.943

Decision concerning the hypotheses:

Because the calculated t is within the critical values, we fail to reject the null hypothesis that there is not a difference in the two stores’ ratings.

Conclusion based on the decision:

Therefore, at the .05 level of significance, the difference in the ratings is not statistically significant.