Statistics: Confidence Problems
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Confidence
Interval Problem Answers AP Statistics Quiz A – Chapter
– Key
A statistics professor asked her students whether or not they were registered to vote. In a sample of
50 of her students (randomly sampled from her 700 students), 35 said they were registered to vote.
1. Find a 95% confidence interval for the true proportion of the professor’s students who were registered to vote. (Make sure to check any necessary conditions and to state a conclusion in the context of the problem.)
We have a random sample of less than 10% of the professor’s students, with 35 expected successes
(registered) and 15 expected failures (not registered), so a Normal model applies.
n
50, pˆ
35
50
0.70, qˆ 1 pˆ
Our 95% confidence interval is: pˆ r z SE pˆ 0.70 r 1.96 0.065
0.30 , so SE pˆ
0.70 r 0.127
ˆˆ pq n
0.70 0.30
50
0.065
0.573 to 0.827
We are 95% confident that between 57.3% and 82.7% of the professor’s students are registered to vote.
2. Explain what 95% confidence means in this context.
If many random samples were taken, 95% of the confidence intervals produced would contain the actual percentage of the professor’s students who are registered to vote.
3. What is the probability that the true proportion of the professor’s students who were registered to vote is in your confidence interval?
There is no probability involved—once the interval is constructed, the true proportion of the professor’s students who were registered to vote is in the interval or it is not.
4. According to a September 2004 Gallup poll, about 73% of 18- to 29-year-olds said that they were registered to vote. Does the 73% figure from Gallup seem reasonable for the professor’s students? Explain.
The 73% figure from Gallup seems reasonable since 73% lies in our confidence interval.
5. If the professor only knew the information from the September 2004 Gallup poll and wanted to estimate the percentage of her students who were registered to vote to
Confidence
Interval Problem Answers AP Statistics Quiz A – Chapter
– Key
A statistics professor asked her students whether or not they were registered to vote. In a sample of
50 of her students (randomly sampled from her 700 students), 35 said they were registered to vote.
1. Find a 95% confidence interval for the true proportion of the professor’s students who were registered to vote. (Make sure to check any necessary conditions and to state a conclusion in the context of the problem.)
We have a random sample of less than 10% of the professor’s students, with 35 expected successes
(registered) and 15 expected failures (not registered), so a Normal model applies.
n
50, pˆ
35
50
0.70, qˆ 1 pˆ
Our 95% confidence interval is: pˆ r z SE pˆ 0.70 r 1.96 0.065
0.30 , so SE pˆ
0.70 r 0.127
ˆˆ pq n
0.70 0.30
50
0.065
0.573 to 0.827
We are 95% confident that between 57.3% and 82.7% of the professor’s students are registered to vote.
2. Explain what 95% confidence means in this context.
If many random samples were taken, 95% of the confidence intervals produced would contain the actual percentage of the professor’s students who are registered to vote.
3. What is the probability that the true proportion of the professor’s students who were registered to vote is in your confidence interval?
There is no probability involved—once the interval is constructed, the true proportion of the professor’s students who were registered to vote is in the interval or it is not.
4. According to a September 2004 Gallup poll, about 73% of 18- to 29-year-olds said that they were registered to vote. Does the 73% figure from Gallup seem reasonable for the professor’s students? Explain.
The 73% figure from Gallup seems reasonable since 73% lies in our confidence interval.
5. If the professor only knew the information from the September 2004 Gallup poll and wanted to estimate the percentage of her students who were registered to vote to