Answer the following items on this page. In these
items, the phrase "margin of error" refers to a 95% confidence interval
or two standard errors. This can be represented as + or - a certain amount.
1. In a college class with 25 students, 12 of whom are
black, the mean on the midterm was 75. The standard deviation was 5. What
is the margin of error for this mean?
Use Formula Four, because the question
is about a mean score. M = 2 * sd / sqrt(N) M =
2 * 5/SQRT(25) = 10/5 = 2.
This is not 2%, it is 2 test score points. The fact that 12 students were
black is irrelevant since the questions does not ask about the scores of
the black students separately.
2. A researcher wants to obtain a margin of error of no
more than 6% in a survey of a county with a population of 300,000. How
large a sample is needed?
Use Formula Three, since the questions
asks how large a sample is needed. We need a 6% margin of error,
which we change to a proportion .06. N = 1/.06 * .06.
= 1/.0036
= 278.
3. 65% of the respondents in a survey of a state with seven million Republican voters voted for Bush, 35% for Gore. There were 625 respondents. What is the margin of error for the percent voting for Bush? Since we are are given a proportion, we use Formula Two. M = 2 * SQRT((p * (1-p))/N). M = 2 * SQRT((.65*.35)/625)) = .03815 = 3.8%
4. What is the margin of error for the Gore vote? It is the same as the previous item since .65 * .35 = .35 * .65. 3.8%
5. In a survey of 1000 voters, 600 were Democrats, 300 Republicans and 100 Libertarian. 85% of the Republicans favored George Bush in the primary. What is the margin of error for this percentage? We use formula Two, since we are given a proportion. In this case, N = 300 since the question is about the Republicans only. M = 2* SQRT((.85*.15)/300))=.041 = 4.1%.
6. A survey is to be conducted of attitudes among white, black and hispanic respondents in Camden County. The population is 300,000. Of this population, 80% is white, 15% is black and 4% is Hispanic. The researcher wants to achieve a 5% margin of error for the estimates for each of the groups. How large a sample is needed? We use formula Three. The margin of error is 5% or .05 which by formula Three requires a sample of 400. We need a 5% margin of error for each of three groups, so the sample required is 3 * 400 = 1200. The proportion white, black and hispanic is irrelevant since we need the same size sample to achieve the same margin of error, regardless of the group's size.
7. In a survey of community residents, the mean income was $18,745 with a standard deviation of $1,345. There are 370,000 residents in the community. 1000 were sampled in the survey. What is the margin of error for this mean score? This is a mean score question, so we use Formula Four. M = 2 * 1345/SQRT(1000) = 2690/31.62 = 85.06 = $85.06. This answer must be in dollars, not in percent.
8. In a survey of Camden County residents, 85% were in favor of a proposal to abolish traffic circles. There are 300,000 county residents, of whom 200 were sampled. What is the margin of error on this percentage? Frmula Two M = 2 * SQRT ((.85*.15)/200) = 5.1%
9. A survey of the tri-county area has 356 respondents,
of whom 82 are black and 45 hispanic. What is the margin of error for statistics
about the opinion of the hispanic residents?
We use Formula one since no percents
are given. N is 45 because we are asked about the Hispanic respondents.
M = 1/SQRT(45) = 14.9%
10. What is the margin of error for statistics based on the responses of all the respondents? The formula is the same, but N = 356. M=1/SQRT(356) = 5.3%
11. In a survey of Rutgers students, 250 white students were sampled and 117 black students. 87% of the students said that Methods and Techniques of Social Research was the most useful class they ever had. What is the margin of error for this percentage? The questions asks about all the students, so N = 250+117= 367. M = 2*SQRT((.87*.13)/367) = 3.5%
12. A study of Rutgers Camden Sociology Department graduates showed that the mean annual salary was $55,000 with a standard deviation of $4500. Three hundred graduates were sampled. What is the margin of error for this statistic? We use Formula Four for mean scores. M = 2 * 4500/sqrt(300) = $519.62. The answer is in dollars.