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Chapter 9: Problem 13

Determine the point estimate of the population mean and margin of error for each confidence interval. Lower bound: \(18,\) upper bound: 24

Short Answer

Expert verified
Point Estimate: 21, Margin of Error: 3

Step by step solution

01

- Identify the Confidence Interval Bounds

Identify the lower and upper bounds of the confidence interval. Given are the lower bound: 18, and the upper bound: 24.
02

- Calculate the Point Estimate of the Population Mean

The point estimate of the population mean is the midpoint of the confidence interval. It can be calculated using the formula: \[ \text{Point Estimate} = \frac{(\text{Lower Bound} + \text{Upper Bound})}{2} \] Substitute the given values: \[ \text{Point Estimate} = \frac{(18 + 24)}{2} = 21 \]
03

- Calculate the Margin of Error

The margin of error is half the width of the confidence interval. It can be calculated using the formula: \[ \text{Margin of Error} = \frac{(\text{Upper Bound} - \text{Lower Bound})}{2} \] Substitute the given values: \[ \text{Margin of Error} = \frac{(24 - 18)}{2} = 3 \]

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Key Concepts

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

Point Estimate
The point estimate is a key concept in statistics. It is used to provide a single value that serves as a best guess or approximation of an unknown population parameter. In this case, we are trying to estimate the population mean.
The point estimate of the population mean is calculated by finding the midpoint of the confidence interval. To do this, add the lower and upper bounds and then divide by two.
For the given exercise, the confidence interval bounds are 18 and 24. So, the point estimate is: \( \text{Point Estimate} = \frac{(18 + 24)}{2} = 21 \)
This calculation gives us the best single estimate for the average value (mean) of the entire population based on the given data.
Margin of Error
Another essential concept is the margin of error. This value tells us how much the point estimate might vary, giving a range of how far off the actual population mean could be from our estimate.
In the given exercise, the margin of error is calculated by taking half of the width of the confidence interval.
The formula is: \( \text{Margin of Error} = \frac{(\text{Upper Bound} - \text{Lower Bound})}{2} \)
Substituting the values: \( \text{Margin of Error} = \frac{(24 - 18)}{2} = 3 \)
This tells us that our point estimate (21) could be 3 units above or below the actual population mean.
Population Mean
The population mean is the average value of a particular characteristic across an entire population. It鈥檚 a fundamental statistical measure.
Due to practical constraints, we often can't calculate the population mean directly, especially in large populations, so we estimate it using sample data and confidence intervals.
In the exercise provided, we cannot know the true population mean directly but estimate it to be around 21 based on the interval provided.
The confidence interval (18 to 24) suggests that we are fairly certain the true mean lies within this range.
Confidence Interval Bounds
Confidence interval bounds indicate the range within which we expect the true population parameter (mean) to fall, given a certain level of confidence.
In our exercise, the lower bound is 18, and the upper bound is 24. These bounds tell us that we are confident that the true population mean lies between these two points.
These bounds are derived from the margin of error and provide a measure of the reliability of our estimate.
The width of this interval (24 - 18 = 6) also gives us an idea of the precision of our estimate 鈥� narrower intervals suggest more precise estimates.
Statistical Computation
Statistical computation involves various mathematical techniques to make sense of data and draw conclusions. In the context of this exercise, it involves calculating the point estimate and margin of error from the given confidence interval.
These computations are guided by established formulas and methodologies to ensure reliable and valid results.
By applying the formulas correctly and using accurate data, we can make informed and reliable estimations about the population parameters, such as the mean in this case.
These calculations are fundamental in many fields, including social sciences, economics, and natural sciences, to analyze and interpret data effectively.

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Most popular questions from this chapter

A researcher wishes to estimate the proportion of households that have broadband Internet access. What size sample should be obtained if she wishes the estimate to be within 0.03 with \(99 \%\) confidence if (a) she uses a 2009 estimate of 0.635 obtained from the National Telecommunications and Information Administration? (b) she does not use any prior estimates?

Determine the point estimate of the population proportion, the margin of error for each confidence interval, and the number of individuals in the sample with the specified characteristic, \(x,\) for the sample size provided. Lower bound: \(0.853,\) upper bound: \(0.871, n=10,732\)

A \(90 \%\) confidence interval for the number of hours that full-time college students sleep during a weekday is lower bound: 7.8 hours and upper bound: 8.8 hours. Which of the following represents a reasonable interpretation of the result? For those that are not reasonable, explain the flaw. (a) \(90 \%\) of full-time college students sleep between 7.8 hours and 8.8 hours. (b) We are \(90 \%\) confident that the mean number of hours of sleep that full- time college students get any day of the week is between 7.8 hours and 8.8 hours. (c) There is a \(90 \%\) probability that the mean hours of sleep that full-time college students get during a weekday is between 7.8 hours and 8.8 hours. (d) We are \(90 \%\) confident that the mean hours of sleep that fulltime college students get during a weekday is between 7.8 hours and 8.8 hours.

A researcher for the U.S. Department of the Treasury wishes to estimate the percentage of Americans who support abolishing the penny. What size sample should be obtained if he wishes the estimate to be within 2 percentage points with \(98 \%\) confidence if (a) he uses a 2006 estimate of \(15 \%\) obtained from a Coinstar National Currency Poll? (b) he does not use any prior estimate?

Determine the critical value \(z_{\alpha / 2}\) that corresponds to the given level of confidence. \(98 \%\)
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