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Chapter 8: Problem 3

News coverage during a recent election projected that a certain candidate would receive \(54.8 \%\) of all votes cast; the projection had a margin of error of \(\pm 3 \%\) a. Give a point estimate for the proportion of all votes the candidate will receive. b. Give an interval estimate for the proportion of all votes the candidate will receive. c. In your own words, state the difference between a point estimate and an interval estimate.

Short Answer

Expert verified
a. Point estimate: 54.8%. b. Interval estimate: [51.8%, 57.8%]. c. Point estimate is a single estimate; interval estimate is a range.

Step by step solution

01

Understanding the Point Estimate

The point estimate is the single value that is the best guess for the proportion of votes the candidate will receive. It is given directly by the projection in the problem statement.
02

Determining the Point Estimate

The point estimate is the projected percentage of votes the candidate is expected to receive, which is given as \(54.8\%\).
03

Understanding the Margin of Error

The margin of error indicates how much the estimated percentage can fluctuate. It is given as an additional range of \(\pm 3\%\) around the point estimate.
04

Calculating the Interval Estimate

The interval estimate is calculated by adding and subtracting the margin of error from the point estimate. This results in a range within which the true proportion is likely to fall. The calculations are as follows:- Lower limit: \(54.8\% - 3\% = 51.8\%\)- Upper limit: \(54.8\% + 3\% = 57.8\%\)Therefore, the interval estimate is \([51.8\%, 57.8\%]\).
05

Explaining Point vs Interval Estimates

A point estimate provides a single best guess value for a quantity, such as the candidate's percentage of votes. An interval estimate, on the other hand, provides a range in which the true value is likely to fall, accounting for potential errors and variability in the estimation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point Estimate
In statistics, when we talk about estimating a population parameter, we often refer to a point estimate. This is a single value that serves as our best "pinpoint" prediction based on sample data. In the context of the election example, the point estimate tells us about the percentage of votes a candidate is predicted to receive based solely on current data.

It's important to understand that a point estimate provides just one best guess at the true parameter value but doesn't account for possible variability or uncertainty.
  • It is incredibly precise; however, that does not mean it is always accurate, since it does not reflect the possibility of error.
  • In the election example, the point estimate provided is 54.8%. This is the straightforward prediction of the vote percentage for the candidate based on the data collected or projected.
In essence, the point estimate is what we think the true proportion is, based on our sample, without any frills or safety nets.
Interval Estimate
An interval estimate extends the concept of a point estimate by offering a range within which the true population parameter is expected to fall. This approach acknowledges potential errors in our estimation process and accounts for variability.

Unlike the definitive nature of a point estimate, interval estimates provide a segment on a line rather than a specific point.
  • The interval estimate is calculated by taking the point estimate and adding and subtracting a value known as the margin of error.
  • In the given election scenario, the point estimate of 54.8% is expanded into an interval by considering a margin of error of 卤3%.
  • Thus, the interval estimate becomes:
    • Lower limit: 54.8% - 3% = 51.8%
    • Upper limit: 54.8% + 3% = 57.8%
    This suggests the candidate's true vote share is likely to fall between 51.8% and 57.8%.
With interval estimates, we introduce a level of confidence regarding our estimate, acknowledging that there is a range of possible true values that the estimated parameter may realistically reflect.
Margin of Error
The margin of error is a vital component of statistical estimation, especially in polling and survey results. It is a measure of the amount by which the estimate might deviate from the true population value due to random sampling variability.

The margin of error provides context to any point estimate by specifying the bounds within which the actual figure likely falls.
  • In essence, it reflects the level of precision of our estimate, acting as a buffer for uncertainty in our projection.
  • To calculate an interval estimate, you use the point estimate and adjust it by the margin of error: one adjustment up and one down.
  • In the election example, with a point estimate of 54.8% and a margin of error of 卤3%, the range of vote percentage is consequently adjusted accordingly.
Understanding and interpreting the margin of error allows us to appreciate the potential variance in survey or polling estimates and to provide a degree of reliability in the results we present.

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Most popular questions from this chapter

Length of hospital stay for childbirth Data was collected from the records of 2962 patients admitted to a hospital in 2015 to estimate the mean length of stay for childbirth. It was observed that the sample mean was 2.372 days and the margin of error for estimating the population mean is 0.029 at a confidence level of \(95 \% .\) Explain the meaning of the last sentence, showing what it suggests about the \(95 \%\) confidence interval. Find the sample standard deviation.

Effect of confidence level Find the margin of error for estimating the population mean when the sample standard deviation equals 100 for a sample size of \(25,\) using confidence levels (i) \(95 \%\) and (ii) \(99 \%\). What is the effect of the choice of confidence level? (You can use Table \(\mathrm{B}\) in the back to find the appropriate \(t\) -scores.)

In a clinical study (the same as mentioned in Example 4 ), 3900 subjects were vaccinated with a vaccine manufactured by growing cells in fertilized chicken eggs. Over a period of roughly 28 weeks, 24 of these subjects developed the flu. a. Find the point estimate of the population proportion that were vaccinated with the vaccine but still developed the flu. b. Find the standard error of this estimate. c. Find the margin of error for a \(95 \%\) confidence interval. d. Construct the \(95 \%\) confidence interval for the population proportion. Interpret the interval. e. Can you conclude that fewer than \(1 \%\) of all people vaccinated with the vaccine will develop the flu? Explain by using results from part d.

A survey was conducted by Mercom Communications India in 2014. Out of 1700 respondents, 1479 stated that they support subsidies for solar power over other sources. a. Estimate the population proportion who supported subsidies for solar power over other sources of energy. b. Find the margin of error for a \(95 \%\) confidence interval for this estimate. c. Find a \(95 \%\) confidence interval for that proportion and interpret. d. State and check the assumptions needed for the interval in part \(c\) to be valid.

Movie recommendation In a quick poll at the exit of a movie theater, 8 out of 12 randomly polled viewers said they would recommend the movie to their friends. a. Construct an appropriate \(95 \%\) confidence interval for the population proportion. b. Is it plausible that only half of all the viewers will be willing to recommend the film to their friends? Explain.
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