Confidence Intervals Around Percentages



Homework - Confidence Intervals Around Percentages

On November 4, 1996, USAToday released the results of their final political poll before the 1996 presidential election. They reported that 51% favored Clinton and 35% favored Dole. This poll included 758 participants.

1. Calculate the standard error for each of these reported percentages.

2. Calculate the 95% and 99% confidence interval for each of these reported percentages.

3. Explain what each of these confidence interval means.

USAToday reported a margin of error of plus or minus 4 percentage points for this poll.

4. What is the basis for the newspaper’s claim that their poll has a margin of error of plus or minus 4 percentage points?

5. Describe a more accurate way, from a statistical perspective, for the newspaper to report the margin of error for this poll.

On November 6, 2000, USAToday released the results of their final political poll before the 2000 presidential election. They reported that 47% favored Bush and 45% favored Gore. This poll included 2,386 participants.

6. Calculate the standard error for each of these reported percentages.

7. Calculate the 95% and 99% confidence interval for each of these reported percentages.

8. Explain what each of these confidence intervals means.

USAToday reported a margin of error of plus or minus 2 percentage points.

9. What is the basis for the newspaper’s claim that their poll has a margin of error of plus or minus 2 percentage points?

10. Describe a more accurate way, from a statistical perspective, for the newspaper to report the margin of error for this poll.

11. Create a plot that depicts the two 95% confidence intervals you calculated in question two alongside the two 95% confidence intervals you calculated in question 7. Label the y-axis with percentage points. Place the four candidates at different points along x-axis, placing opponents next to each other.

12. Compare and contrast the sets of confidence intervals for the two elections.

13. If you were interpreting these polls on the eve of Election Day, what predictions would you have made about the outcome of the popular vote in each of these elections?

The actual results of the 1996 popular vote showed that Clinton received 49% to Dole’s 41%. The actual results of the 2000 popular vote showed that both Bush and Gore received approximately 49% of the vote.

14. In what respects did the USAToday correctly call the results of each popular vote? Were they wrong in any way?

Sampling Distribution of the Mean

Group Activity

Have one member of the group shuffle a deck of cards. Have another member of the group select one card from the deck. Count an ace as 1, cards 2 through 10 as their face value, a Jack as 11, a Queen as 12, and a King as 13. Record the point value of the card. Replace the card in the deck, reshuffle, and have the same group member select a card again. You may save time by randomly inserting the chosen card back into the deck out of the sight of the person who is selecting the cards. Repeat this process until you have drawn 30 random samples of size 7 from the deck.

1. Calculate the mean and standard deviation of each sample based on these point values.

2. Using the point values given, plot the frequency distribution for the population from which you were sampling using a histogram.

3. Plot the frequency distribution for the first sample you selected using a histogram.

4. Plot the frequency distribution of means across your 30 samples.

5. Plot the theoretical sampling distribution of the mean for this particular sampling situation.

The Binomial Distribution

Group Activity

Have one member of the group shuffle a deck of cards. Have another member of the group select one card from the deck. Record whether the card is red or black. Replace the card in the deck, reshuffle, and have the same group member select a card again. You may save time by randomly inserting the chosen card back into the deck out of the sight of the person who is selecting the cards. Repeat this process until you have drawn 30 random samples of size 8 from the deck.

1. Record the number and percent of red and black cards for each sample.

2. Plot the frequency distribution for the population from which you were sampling using a histogram.

3. Plot the frequency distribution for the first sample you selected using a histogram.

4. Plot the frequency distribution of the number of red cards in each of your 30 samples.

5. Plot the theoretical sampling distribution of the number of red cards for this particular sampling situation.

Sampling Distribution of the Percentage

Group Activity

The attached page shows the results of the USAToday tracking poll for the percentage of those polled who favored Clinton prior to the 1996 presidential election. The results are for the period Sept. 4, 1996 through Nov. 4, 1996. They represent the results of this poll for the sixty-two days immediately preceding the election. The sample size each day was 758. When the first poll was conducted on Sept. 4, 1996 the results showed that 53% favored Clinton and 36% favored Dole. The final poll taken on Nov. 4 showed that 51% favored Clinton and 35% favored Dole. These results remained in a very narrow range throughout the sixty-two days. In addition, this election took place during a time of relative peace, prosperity, and satisfaction. The two candidates, for all their many strengths and weaknesses, were very well known to the American people. No major news events took place during this time period to change the basic pattern of preferences. Therefore, let us assume that these daily polls represent repeated random samples from the same population.

1. Calculate the mean and standard deviation of the daily poll results for each candidate.

2. Plot the frequency distribution of poll results for each candidate using a histogram.

3. Now assume that 52% for Clinton and 35% for Dole are the actual population percentages. Calculate the standard error for the distribution of sample percentages given repeated samples of size 758.

4. Plot the sampling distributions for the conditions given in question 3.

5. Compare the standard deviations of the distribution of poll results to the standard errors you calculated in question 3.

6. What you expect to happen to the frequency distribution of sample percentages form question 2 if you were to continue to replicate over a very large number of trials during which nothing changed about the pattern of preferences in the population?

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