Teacher Material …. Partial Solutions / Examples …. Do Not give to Students
Name Partial Solutions / Examples
Date ______
CONFIDENCE INTERVAL WORKSHEET ( Some answers will vary! )
1)Suppose in a state with a large number of voters that 56 out of 100 randomly surveyed voters favored Proposition 1. This is just a small sample of all the voters. Do you think Proposition 1 passed? YES or NO (circle answer)
Using your intuition, how confident are you that Proposition 1 passed or failed? (circle your answer.)
Reasonably sure!
Not very sure, I would like more information.
I’d bet a small amount of money that I am right.
I’d bet a lot of money that I am right.
2)In the mathematical field of statistics, the term parameter is used for measures of an entire population, while the term statistic is reserved for measures of a sample.
Let P = the entire population proportion having a given characteristic.
Let (p bar) = the sample proportion having a given characteristic. In our example #1, = 56/100 = .56 = 56%. You can see that will vary from sample to sample and will also depend partly on the sample size, n. An advanced theorem from statistics states three things about the values of .
- The values of are normally distributed.
- The mean of this distribution is P.
- The standard deviation of this distribution is:
What is the sample size for the problem in #1?
n = 100What is the standard deviation of the distribution for the problem in #1
s.d. = 0.0496386946....What is the standard deviation rounded to the nearest tenth of a percent?
s.d. = 5.0%3)Since is normally distributed, we can apply the 68% - 95% - 99% rule which says:
- About 68% of the distribution is within 1 standard deviation of the mean.
- About 95% of the distribution is within 2 standard deviations of the mean.
- About 99% of the distribution is within 3 standard deviations of the mean.
For instance, 95% of the time is within 2 standard deviations of it’s true mean , P. Since we use to estimate P, the following equivalent statement is more useful: 95% of the time, P is within 2 standard deviations of . This interval in which P lies 95% of the time is called a 95% confidence interval.
Compute a 95% confidence interval (to the nearest tenth of a percent) for our example in problem #1.
95% CONFIDENCE INTERVAL TABLE- 2*(standard deviation) / < P < / + 2*(standard deviation)
- 2* / < P < / + 2*
46.1% / < P < / 65.9%
4)State what this means in words by completing the following sentences!
I am confident that 95% of the time, the true proportion of voters that actually voted for Proposition 1 is between 46.1% and 65.9%Other samples of 100 voters will yield other 95% confidence intervals. Most of these confidence intervals (about 95% of them) will capture P, but a few of them (about 5%) will not. For this reason, we can be 95% confident that the unknown value of P is between 46.1% and 65.9%.
5)The 95% confidence interval we just computed is rather wide and does not pinpoint P to any great extent. (In fact, we can not even tell whether a majority voted for Proposition 1.) Now, how confident are you that Proposition 1 passed or failed? (circle your answer.)
Reasonably sure!
Not very sure, I would like more information.
I’d bet a small amount of money that I am right.
I’d bet a lot of money that I am right.
6)Our next example shows that we can obtain a narrower confidence interval by taking a larger sample.
Example 2: Suppose in a state with a large number of voters that 560 out of 1000 randomly surveyed voters favored Proposition 1.
What is the sample size for example #2?
n = 1000What is the standard deviation of the distribution for example#2
s.d. = 0.0156971335...What is the standard deviation rounded to the nearest tenth of a percent?
s.d. = 1.6%7)Compute a 95% confidence interval (to the nearest tenth %) for example #2.
95% CONFIDENCE INTERVAL TABLE – Example #2- 2*(standard deviation) / < P < / + 2*(standard deviation)
- 2* / < P < / + 2*
52.9% / < P < / 59.1%
8)State what this means in words!
I am confident that 95% of the time, the true proportion of voters that actually voted for Proposition 1 is between 52.9% and 59.1%9)Notice that the sample size of 1000 gives a much narrower confidence interval than the sample size of 100. In fact, with the larger sample, we can be quite confident (about 95% of the time anyway), that a majority of the voters favored Proposition 1, since the smaller endpoint of the samples 95% confidence interval, 0.529 is greater than one-half. Bear in mind, however, that the larger sample may be more costly and time consuming than the smaller one. Now, how confident are you that Proposition 1 passed or failed? (circle your answer.)
Reasonably sure!
Not very sure, I would like more information.
I’d bet a small amount of money that I am right.
I’d bet a lot of money that I am right.
10)Sometimes a 95% confidence interval is not enough. For example, in testing new medical drugs or procedures, a 99% confidence interval may be required before the new drug or procedure is approved for general use. A 99% confidence interval is based on P being within 3 standard deviations of . Compute a 99% confidence interval (to the nearest tenth %) for example #2.
99% CONFIDENCE INTERVAL TABLE – Example #2- 3*(standard deviation) / < P < / + 3*(standard deviation)
- 3* / < P < / + 3*
51.3% / < P < / 60.7%
11)State what this means in words!
I am confident that 99% of the time, the true proportion of voters that actually voted for Proposition 1 is between 51.3% and 60.7%12)Is the 99% confidence interval wider or narrower than the 95% confidence interval? (circle your answer.)
WIDER / NARROWER
13)State why!
Instead of going 2 standard deviations to the left and right of the sample mean, I am now going 3 standard deviations to the left and right of the sample mean.There is a program on the TI83 called CONFIDE that will compute a 95% confidence interval and a 99% confidence interval. Familiarize yourself with the program by verifying your results for questions 3, 7, and 10.
14)By either using the program CONFIDE or a calculator, compute 95% and 99% confidence intervals for the following situation:
Number in favor = 510 Number in sample = 1000
(This time leave your answers as decimals to 3 places)
95% CONFIDENCE INTERVAL.478 / < P < / .542
99% CONFIDENCE INTERVAL
.463 / < P < / .557
Although 51% (in our sample) voted in favor, would you tell the newspaper to go ahead and print the paper with the headline that this proposition passed? YES or NO (circle your answer).
15)By either using the program CONFIDE or a calculator, compute 95% and 99% confidence intervals for the following situation:
Number in favor = 5,100 Number in sample = 10,000
(Leave your answers as decimals to 3 places.)
95% CONFIDENCE INTERVAL.500 / < P < / .520
99% CONFIDENCE INTERVAL
.495 / < P < / .525
Although 51% (in our sample) voted in favor, would you tell the newspaper to go ahead and print the paper with the headline that this proposition passed? YES or NO (circle your answer).
16)How large of a sample (within reason) would you want before you told your boss to go ahead and print the paper with the headline that this proposition passed. (For this question, we are only considering a sample that produced a 51% in favor proportion).
I would not be comfortable telling my boss to go ahead until I had a sample size of at least 25,000 voters. The 99% confidence interval for this sample size (with a 51% in favor proportion is:.50052 / < P < / .51948
And the number of voters that would have had to voted in favor is: 12,750
17)Maybe this vote is too close too call! Comment on what you would tell the newspaper to do in this situation.
Answers will vary. Perhaps the newspaper should print the headline as “VOTE IS TOO CLOSE TO CALL”, because someone will likely ask for a recount.Also, a sample size of 25,000 or more may be too costly or time consuming
to be feasible for the newspaper.
18)Write a summary commenting on what you have learned about confidence intervals in polls and surveys and how sample size affects the intervals.
ANSWERS WILL VARY!1