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Chapter 8: Problem 2

Explain the meaning of a point estimate and an interval estimate.

Short Answer

Expert verified
A point estimate is a single statistic that estimates a population parameter, acting as the best guess of the true value. However, an interval estimate is a range of values that is likely to include the population parameter, offering more information and an attached level of confidence.

Step by step solution

01

Definition of Point Estimate

A point estimate of a population parameter is a single value of a statistic. For example, the sample mean, x, is a point estimate of the population mean, \(\mu\). Essentially, it is the best guess of the true parameter value, based on sample data.
02

Definition of Interval Estimate

An interval estimate is an estimate of a population parameter that provides an interval suspected to contain the value of the parameter. More technically, it contains a range of values instead of a single point estimate. Let's say your interval estimate of the population mean is \([18, 22]\). According to this estimate, the population mean is likely to fall anywhere between 18 and 22.
03

Contrasting Both Terms

While a point estimate involves a single statistic to estimate a parameter, an interval estimate, by contrast, gives you a range of values in which the parameter is likely to fall - providing you with more information about the parameter you're trying to estimate. Notably, an interval estimation comes with a level of confidence attached - so, you could say our earlier example of an interval estimate \([18, 22]\) is with 95% confidence, meaning we're 95% certain the actual population mean lies within that range.

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Key Concepts

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

Point Estimate
A point estimate is a single, specific value used to approximate a population parameter. Think of it as a snapshot or a minimalistic summary of the entire data collected. For example, if you want to estimate the average height of students in a school, you might calculate the average (mean) height from a sample of students. This sample mean \(x\) serves as a point estimate for the real average height \( \mu \) of all students in the school.
  • It's concise: a single number is easy to understand and communicate.
  • It's direct: it gives an immediate approximation of the parameter.
However, one should remember that a point estimate does not provide any insight into the variability or uncertainty of the estimation.
Interval Estimate
An interval estimate provides a range of values within which the true population parameter is expected to lie. It offers a broader perspective than a point estimate by giving you a spectrum in which you expect the real value to be. Let's say we estimate the average height again, but this time our interval estimation is \[ (150 \, \text{cm}, 170 \, \text{cm}) \]. This range suggests that the actual average height could be any value within those boundaries.
  • More informative: Offers a better understanding by considering estimation uncertainty.
  • Safety net: Accounts for sampling error to some extent by providing a range instead of a single value.
Despite its broader information, the interval still needs context for its "confidence" level.
Confidence Interval
Confidence intervals are a type of interval estimate that is enriched with a confidence level. This confidence level typically expresses how certain we are that our interval contains the true population parameter. For instance, you might say your confidence interval for students' average height is \( (150 \, \text{cm}, 170 \, \text{cm}) \) with a 95% confidence level. This means you are 95% confident that the true mean height lies within this range.
  • Credibility: Lends statistical trustworthiness to interval estimates by quantifying certainty.
  • Flexibility: Varying the confidence level (e.g., from 90% to 99%) modifies the range width, generally increasing certainty but also broadening the interval.
Understanding confidence intervals helps in making more informed decisions based on statistical data.
Population Parameter
The term population parameter refers to a characteristic or measure of an entire population. Unlike a sample statistic, it is the "true" value, although it is often unknown and must be estimated. Key population parameters include the mean \( \mu \), variance \( \sigma^2 \), and standard deviation \( \sigma \).
  • It is an assumed priority: The ultimate goal in statistics is often to determine or closely estimate the value of population parameters.
  • Foundation for Estimates: Population parameters serve as the basis for both point and interval estimates.
Being able to determine or accurately estimate population parameters allows statisticians and researchers to better understand the entire dataset's behavior, rather than just a sample of it.

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Most popular questions from this chapter

A random sample of 34 participants in a Zumba dance class had their heart rates measured before and after a moderate 10 -minute workout. The following data correspond to the increase in each individual's heart rate (in beats per minute): \(\begin{array}{llllllllllll}59 & 70 & 57 & 42 & 57 & 59 & 41 & 54 & 44 & 36 & 59 & 61 \\ 52 & 42 & 41 & 32 & 60 & 54 & 52 & 53 & 51 & 47 & 62 & 62 \\ 44 & 69 & 50 & 37 & 50 & 54 & 48 & 52 & 61 & 45 & & \end{array}\) a. What is the point estimate of the corresponding population mean? b. Make a \(98 \%\) confidence interval for the average increase in a person's heart rate after a moderate 10 -minute Zumba workout.

According to the 2015 Physician Compensation Report by Medscape (a subsidiary of WebMD), American orthopedists earned an average of \(\$ 421,000\) in 2014 . Suppose that this mean is based on a random sample of 200 American orthopaedists, and the standard deviation for this sample is \(\$ 90,000\). Make a \(90 \%\) confidence interval for the population mean \(\mu\).

The standard deviation for a population is \(\sigma=14.8\). A random sample of 25 observations selected from this population gave a mean equal to \(143.72\). The population is known to have a normal distribution. a. Make a \(99 \%\) confidence interval for \(\mu\). b. Construct a \(95 \%\) confidence interval for \(\mu\). c. Determine a \(90 \%\) confidence interval for \(\mu\). d. Does the width of the confidence intervals constructed in parts a through \(\mathrm{c}\) decrease as the confidence level decreases? Explain your answer.

Briefly explain the meaning of the degrees of freedom for a \(f\) distribution. Give one example.

Calculating a confidence interval for the proportion requires a minimum sample size. Calculate a confidence interval, using any confidence level of \(90 \%\) or higher, for the population proportion for each of the following. a. \(n=200\) and \(\hat{p}=.01\) b. \(n=160\) and \(\hat{p}=.9875\) Explain why these confidence intervals reveal a problem when the conditions for using the normal approximation do not hold.
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