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Chapter 8: Problem 1

Briefly explain the meaning of an estimator and an estimate.

Short Answer

Expert verified
In statistics, an estimator is a formula or procedure used to derive estimates based on sample data. An estimate is the actual numerical value derived from applying an estimator to the sample data.

Step by step solution

01

Explanation of Estimator

An estimator refers to a statistical formula or a function used to provide estimates of population parameters based on sample data. It is a rule for calculating an estimate of a given quantity based on observed data. An estimator represents a population parameter using a formula that combines the measurements contained in a sample. Technically, we can say an estimator is a function of the sample. For example, the sample mean is an estimator of the population mean.
02

Explanation of Estimate

On the other hand, an estimate is the numerical value generated by the estimator. It is the outcome of the function of the sample data (the estimator). Once we plug our sample data into an estimator, we get our estimate. If the sample mean (an estimator) computes to be '5', then '5' is the estimate. An estimate aims to infer a population characteristic from sample data.

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Key Concepts

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

Population Parameters
Population parameters are key characteristics that describe a population. These could be things like the population mean (average), variance, or proportion. Imagine we want to know the average height of all the students in a school.

The true average height of the entire student body is a population parameter. However, it's often impractical to measure every single student.
  • Population parameters are often unknown and fixed values.
  • Parameters provide comprehensive insights into population characteristics, but require complete data collection.
  • Examples include the population mean (\( \mu \)), population variance (\( \sigma^2 \)), and population proportion (\( P \)).
Understanding population parameters is the goal, but it's usually achieved through estimating them using sample data.
Sample Data
Sample data is a smaller, manageable version of a larger group or population. Instead of analyzing an entire population, statisticians collect data from a select portion and use it to make inferences about the population.

It's a practical way to gather information, especially when the population is large. Take enough samples, and you'll get a fairly reliable snapshot of the whole group.
  • Sample data consist of observations drawn from a larger population.
  • It should be representative, meaning it accurately reflects the population.
  • Key benefits include saving time and resources.
From this data, we apply statistical methods to guess at the broader population trends.
Statistical Formula
Statistical formulas are the tools we use to turn sample data into useful information about a population. They're the basis of what statisticians call an estimator.

For example, if you want to estimate the average height of students based on a sample, you'd use a statistical formula to compute the sample mean.
  • Statistical formulas apply mathematical operations to sample data.
  • These formulas often involve sums, means, ratios, and variances.
  • They help in deriving estimates for unknown population parameters.
Using statistical formulas wisely ensures that estimates are as accurate and unbiased as possible, given the data.
Function of the Sample
The function of the sample refers to how we use sample data to create estimates of population parameters. An estimator operates as a function of the sample because it's a rule that assigns a number (the estimate) to the data collected in a sample.

The sample mean, for example, is a function of the sample because it takes all the individual data points in the sample, adds them together, and divides by the number of observations.
  • An estimator transforms sample data into estimates.
  • Functions of the sample are foundational in creating meaningful statistical analysis.
  • Being a function, an estimator can handle varying sample sizes and data sets.
By understanding the function of the sample, we can better comprehend how estimates provide a window into the larger population's traits.

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Most popular questions from this chapter

A group of veterinarians wants to test a new canine vaccine for Lyme disease. (Lyme disease is transmitted by the bite of an infected deer tick.) In an area that has a high incidence of Lyme disease, 100 dogs are randomly selected (with their owners' permission) to receive the vaccine. Over a 12 -month period, these dogs are periodically examined by veterinarians for symptoms of Lyme disease. At the end of 12 months, 10 of these 100 dogs are diagnosed with the disease. During the same 12 -month period, \(18 \%\) of the unvaccinated dogs in the area have been found to have Lyme disease. Let \(p\) be the proportion of all potential vaccinated dogs who would contract Lyme disease in this area. a. Find a \(95 \%\) confidence interval for \(p\). b. Does \(18 \%\) lie within your confidence interval of part a? Does this suggest the vaccine might or might not be effective to some degree? c. Write a brief critique of this experiment, pointing out anything that may have distorted the results or conclusions.

a. How large a sample should be selected so that the margin of error of estimate for a \(99 \%\) confidence interval for \(p\) is \(.035\) when the value of the sample proportion obtained from a preliminary sample is \(.29 ?\) b. Find the most conservative sample size that will produce the margin of error for a \(99 \%\) confidence interval for \(p\) equal to \(.035\).

A couple considering the purchase of a new home would like to estimate the average number of cars that go past the location per day. The couple guesses that the number of cars passing this location per day has a population standard deviation of 170 . a. On how many randomly selected days should the number of cars passing the location be observed so that the couple can be \(99 \%\) certain the estimate will be within 100 cars of the true average? b. Suppose the couple finds out that the population standard deviation of the number of cars passing the location per day is not 170 but is actually 272 . If they have already taken a sample of the size computed in part a, what confidence does the couple have that their point estimate is within 100 cars of the true average? c. If the couple has already taken a sample of the size computed in part a and later finds out that the population standard deviation of the number of cars passing the location per day is actually 130, they can be \(99 \%\) confident their point estimate is within how many cars of the true average?

You want to estimate the proportion of students at your college who hold off- campus (part-time or fulltime) jobs. Briefly explain how you will make such an estimate. Collect data from 40 students at your college on whether or not they hold off-campus jobs. Then calculate the proportion of students in this sample who hold off-campus jobs. Using this information, estimate the population proportion. Select your own confidence level.

In a Time/Money Magazine poll of Americans of age 18 years and older, \(65 \%\) agreed with the statement, "We are less sure our children will achieve the American Dream" (Time, October 10,2011 ). Assume that this poll was based on a random sample of 1600 Americans. a. Construct a \(95 \%\) confidence interval for the proportion of all Americans of age 18 years and older who will agree with the aforementioned statement. b. Explain why we need to construct a confidence interval. Why can we not simply say that \(65 \%\) of all Americans of age 18 years and older agree with the aforementioned statement?
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