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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

Sleep apnea is a disorder in which you have one or more pauses in breathing or shallow breaths while you sleep. In a cross-sectional study of 320 individuals who suffer from sleep apnea, it was found that 192 had gum disease. Note: In the general population, about \(17.5 \%\) of individuals have gum disease. (a) What does it mean for this study to be cross-sectional? (b) What is the variable of interest in this study? Is it qualitative or quantitative? Explain. (c) Estimate the proportion of individuals who suffer from sleep apnea who have gum disease with \(95 \%\) confidence. Interpret your result.

Find the critical values \(\chi_{1-\alpha / 2}^{2}\) and \(\chi_{\alpha / 2}^{2}\) for the given level of confidence and sample size. \(98 \%\) confidence, \(n=23\)

Construct the appropriate confidence interval. A simple random sample of size \(n=12\) is drawn from a population that is normally distributed. The sample variance is found to be \(s^{2}=23.7\). Construct a \(90 \%\) confidence interval for the population variance.

A USA Today/Gallup poll asked 1006 adult Americans how much it would bother them to stay in a room on the 13 th floor of a hotel. Interestingly, \(13 \%\) said it would bother them. The margin of error was 3 percentage points with \(95 \%\) confidence. Which of the following represents a reasonable interpretation of the survey results? For those not reasonable, explain the flaw. (a) We are \(95 \%\) confident that the proportion of adult Americans who would be bothered to stay in a room on the 13th floor is between 0.10 and 0.16 . (b) We are between \(92 \%\) and \(98 \%\) confident that \(13 \%\) of adult Americans would be bothered to stay in a room on the 13th floor. (c) In \(95 \%\) of samples of adult Americans, the proportion who would be bothered to stay in a room on the 13 th floor is between 0.10 and 0.16 . (d) We are \(95 \%\) confident that \(13 \%\) of adult Americans would be bothered to stay in a room on the 13 th floor.

A random sample of 1003 adult Americans was asked, "Do you pretty much think televisions are a necessity or a luxury you could do without?" Of the 1003 adults surveyed, 521 indicated that televisions are a luxury they could do without (a) Obtain a point estimate for the population proportion of adult Americans who believe that televisions are a luxury they could do without. (b) Verify that the requirements for constructing a confidence interval about \(p\) are satisfied. (c) Construct and interpret a \(95 \%\) confidence interval for the population proportion of adult Americans who believe that televisions are a luxury they could do without. (d) Is it possible that a supermajority (more than \(60 \%\) ) of adult Americans believe that television is a luxury they could do without? Is it likely? (e) Use the results of part (c) to construct a \(95 \%\) confidence interval for the population proportion of adult Americans who believe that televisions are a necessity.
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