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Chapter 15: Problem 2

Florida Voters. Florida has played a key role in recent presidential elections. Voter registration records in September 2019 show that \(37 \%\) of Florida voters are registered as Democrats and \(\mathbf{3 5} \%\) as Republicans. (Most of the others did not choose a party.) To test a random digit dialing device that you plan to use to poll voters for the 2020 presidential elections, you use it to call 250 randomly chosen residential telephones in Florida. Of the registered voters contacted, \(35 \%\) are registered Democrats. Is each of the boldface numbers a parameter or a statistic?

Short Answer

Expert verified
37% is a parameter; 35% is a statistic.

Step by step solution

01

Define Parameter

A parameter is a number that describes a characteristic of a population. In the context of this problem, a parameter reflects the true proportion of all Florida voters who are registered Democrats or Republicans as seen in the voter registration records.
02

Define Statistic

A statistic is a number that describes a characteristic of a sample drawn from a population. Here, the statistic would reflect the proportion of registered voters among the 250 who were reached by the random digit dialing device.
03

Identify Parameters

The boldfaced numbers in the problem are 35% and 37%. The number 37% represents the proportion of registered Democrats as reported in voter registration records. This describes a characteristic of the population of all Florida voters, thus 37% is a parameter.
04

Identify Statistics

The number 35% describes the proportion of registered Democrats in the sample of 250 contacted registered voters. Since this figure pertains to a sample rather than the entire population, 35% is a statistic.

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Key Concepts

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

Parameter
In the world of statistics, a parameter is a crucial concept. It is a number that gives us precise information about a population.
  • Think of a population as the entire group we are interested in studying. For example, in the Florida voters' scenario, this would include every registered voter in Florida.
  • The parameter represents a feature of this population. In our example, 37% of Florida voters are registered as Democrats, based on registration records. This 37% is a parameter because it tells us about the whole group, not just a part of it.
Parameters are usually fixed and unchanging for a particular population, making them your ultimate target for understanding the entire demographic at once. Understanding parameters helps in making predictions and generalizations about the entire population.
Statistic
A statistic, unlike a parameter, provides information about a sample of a population rather than the whole group. To illustrate:
  • A sample is a smaller, manageable segment that is randomly selected from the larger population.
  • In the exercise about Florida voters, when 250 voters were sampled through random digit dialing, the number 35% is a statistic. It tells us the proportion of registered Democrats in this sample.
Statistics can vary because they depend on which sample is taken. Hence, while parameters are fixed (for the population and the set conditions), statistics can change. This variability is why statistics are often used to infer or estimate parameters, helping us understand the broader population with more manageable data.
Population
The term "population" in statistics refers to the entire group that you are interested in or the whole set of elements you seek to analyze.
  • It could be people, events, measurements, etc.
  • In the exercise example, the population would be all registered voters in Florida as of September 2019.
Understanding populations is essential because they are the target of our interest when determining parameters or making predictions and generalizations. However, due to their vast size, it is often impractical to study the entire population directly, which is why samples are used to gather data efficiently.
Sample
A sample is a smaller, but representative part of a larger population. Choosing a proper sample is a powerful technique in statistics as it manages the unfeasibility of studying entire populations.
  • In the exercise, a sample of 250 Florida voters was used to gather data, thanks to a random digit dialing device.
  • This process helps in estimating the parameters of the population from which the sample was drawn.
The sample needs to be random and unbiased to ensure it accurately reflects the population's characteristics. If properly done, conclusions drawn from the sample can confidently relate to the entire population, thereby simplifying the study of large groups.

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Most popular questions from this chapter

A newborn 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 who had been born with ELBW. Their mean weight at birth was \(x=810\) 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 \(x\) will be equal to \(\mu\). b. as we take larger and larger samples from this population, \(x\) will get closer and closer to \(\mu\). c. in many samples from this population, the many values of \(x\) will have a distribution that is close to Normal.

Detecting the Emerald Ash Borer. The emerald ash borer is a serious threat to ash trees. A state agriculture department places traps throughout the state to detect the emerald ash borer. When traps are checked periodically, the mean number of ash borers trapped is only \(2.2\), but some traps have many ash borers. The distribution of ash borer counts is finite and strongly skewed, with standard deviation \(3.9 .\) a. What are the mean and standard deviation of the average number of ash borers \(x\) in 50 traps? b. Use the central limit theorem to find the probability that the average number of ash borers in 50 traps is greater than 3.0.

The Law of Large Numbers Made Visible. Roll two balanced dice and count the total spots on the up-faces. The probability model appears in Example 12.5 (page 277). You can see that this distribution is symmetric with 7 as its center, so it's no surprise that the mean is \(\mu=7\). This is the population mean for the idealized population that contains the results of rolling two dice forever. The law of large numbers says that the average \(x\) from a finite number of rolls tends to get closer and closer to 7 as we do more and more rolls. a. Click "More dice" once in the Law of Large Numbers applet to get two dice. Click "Show \(\mu_{x}\) " to see the mean 7 on the graph. Leaving the number of rolls at 1 , click "Roll dice" three times, recording each roll. How many spots did each roll produce? What is the average for the three rolls? You see that the graph displays at each point the average number of spots for all rolls up to the last one. This is exactly like Ejgure 15.1. b. Click "Reset" to start over. Set the number of rolls to 100 and click "Roll dice." The applet rolls the two dice 100 times. The graph shows how the average count of spots changes as we make more rolls. That is, the graph shows \(x\) as we continue to roll the dice. Sketch (or print out) the final graph. c. Repeat your work from part (b). Click "Reset" to start over, then roll two dice 100 times. Make a sketch of the final graph of the mean \(x\) against the number of rolls. Your two graphs will often look very different. What they have in common is that the average eventually gets close to the population mean \(\mu=7\). The law of large numbers says that this will always happen if you keep on rolling the dice.

Guns in School. Researchers surveyed 14,765 American high school students (grades 9-12) and found that \(27.3 \%\) of those surveyed were in grade 9 . The percentage of all American high school students who are are in grade 9 is \(\mathbf{2 6 . 5} \%\). The percentage of those surveyed who were in grade 9 and had carried a gun to school was \(\mathbf{4 . 4 \%}\). Is each of the boldface numbers a parameter or a statistic?

Daily Activity. It appears that people who are mildly obese are less active than leaner people. One study looked at the average number of minutes per day that people spend standing or walking. I Among mildly obese people, the mean number of minutes of daily activity (standing or walking) is approximately Normally distributed with mean 373 minutes and standard deviation 67 minutes. The mean number of minutes of daily activity for lean people is approximately Normally distributed with mean 526 minutes and standard deviation 107 minutes. A researcher records the minutes of activity for an SRS of five mildly obese people and an SRS of five lean people. a. What is the probability that the mean number of minutes of daily activity of the five mildly obese people exceeds 420 minutes? b. What is the probability that the mean number of minutes of daily activity of the five lean people exceeds 420 minutes?
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