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Chapter 15: Problem 17

A post-election poll of Canadian adults who were registered voters found that \(77 \%\) said they voted in the Dctober 2015 elections. Election records show that \(68.3 \%\) of registered voters voted in the election. The boldface number is a (a) sampling distribution. (b) statistic. (c) parameter.

Short Answer

Expert verified
The boldface number, 68.3%, is a parameter.

Step by step solution

01

Understand the Terms

A *parameter* is a numerical value summarizing a characteristic of a population. A *statistic* is a numerical value summarizing a characteristic of a sample. *Sampling distribution* refers to the probability distribution of a given statistic based on a random sample.
02

Identify the Parameter

The election records, which report that 68.3% of registered voters participated in the election, represent a *parameter*. This is because it reflects the actual behavior of the entire population of registered voters.
03

Define the Statistic

The survey result that 77% of Canadian adults said they voted is a *statistic*. This number is based on a sample of voters who participated in the poll, not the entire population.
04

Compare the Number in Question

The boldface number in the question, 68.3%, is based on election records as opposed to being calculated from a sample survey. Given this, it is the true participation rate of the population and thus a *parameter*.

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Key Concepts

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

Understanding Parameters in Statistics
In statistics, a parameter is a special term that describes a numerical characteristic of a population. Imagine you have a large group of people, like all the registered voters in Canada. If you wanted to find out how many of them voted in elections, that percentage would be a parameter. The key here is that parameters relate to the entire group鈥攐ur whole population.
For example, in the Canadian election scenario provided, 68.3% of registered voters actually voted. This percentage is a parameter because it represents the behavior of every registered voter, not just a portion.
Parameters are like the ultimate goal in statistics; we often try to estimate them using data and calculations.
Defining Statistics in the Context of Samples
In contrast to parameters, a statistic refers to a number that describes a sample, which is a smaller group selected from a population. It is not always feasible to gather data from every individual. So instead, statisticians often look at a smaller group鈥攐r sample鈥攖o make inferences about the wider population.
Back to our Canadian voter example, the 77% mentioned in the problem is a statistic, because this number comes from a survey of a sample of voters. These voters are only a portion of all Canadian adults. It is a "snapshot" of the opinions of this sample, not the entire registered voter population.
  • Statistics vary depending on the sample.
  • They help us estimate parameters.
Understanding this difference is vital in how we interpret data from surveys and studies.
Exploring Sampling Distributions
Sampling distributions can seem complex at first, but they're all about understanding variation in statistics. When you repeatedly take samples from a population, the values of a statistic will change from sample to sample. A sampling distribution is essentially the collection of these values.
Imagine you take 100 different samples of voters and calculate the percentage of voters in each. The result is a variety of percentages. The distribution of these percentages forms what we call a sampling distribution.
  • It shows all possible values of a statistic.
  • Provides insights into the likelihood of different outcomes.
This concept helps us understand "spread" and "centrality" in statistics, letting us predict how a statistic may behave across different samples. It's a cornerstone in estimating the reliability of statistics.

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Most popular questions from this chapter

What Does the Central Limit Theorem Say? Asked what the central limit theorem says, a student replies, "As you take larger and larger samples from a population, the histogram of the sample values looks more and more Nonal." Is the student right? Explain your answer.

The number of hours a battery lasts before failing varies from battery to battery. The distribution of failure times follows an exponential distribution (see Example \(15.7\) ), which is strongly skewed to the right. The central limit theorem says that (a) as we look at more and more batteries, their average failure time gets ever closer to the mean \(\mu\) for all batteries of this type. (b) the average failure time of a large number of batteries has a distribution of the same shape (strongly skewed) as the distribution for individual batteries. (c) the average failure time of a large number of batteries has a distribution that is close to Normal.

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Scores on the Critical Reading part of the SAT exam in a recent year were roughly Normal with mean 495 and standard deviation 118 . You choose an SRS of 100 students and average their SAT Critical Reading scores. If you do this many times, the mean of the average scores you get will be close to (a) 495 . (b) \(495 / 100==.4 .95\) (c) \(495 / 100=49.5^{495 / \sqrt{100}}=49.5\).

A newbom baby has extremely low birth weight (ELBW) if it weighs less than 1000 grams. A study of the health of such children in later years examined a random sample of 219 children. Their mean weight at birth was \(x^{-}=810\) grams \(\bar{x}=810\) grams. This sample mean is an unbiased estimator of the mean weight \(\mu\) in the population of all ELBW babies. This means that (a) in many samples from this population, the mean of the many values of \(x^{-} \bar{x}\) will be equal to \(\mu\). (b) as we take larger and larger samples from this population, \(x^{-} x\) will get closer and closer to \(\mu\). (c) in many samples from this population, the many values of \(x^{-} \bar{x}\) will have a distribution that is close to Normal.
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