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Chapter 15: Problem 19

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) \(495 / 100=49.5^{495 / \sqrt{100}}=49.5\).

Short Answer

Expert verified
The mean of the average scores is 495. Option (a) is correct.

Step by step solution

01

Understanding the Problem

We are given that the SAT scores are normally distributed with a mean \( \mu = 495 \) and standard deviation \( \sigma = 118 \). We need to find the mean of the average scores when an SRS of 100 students is chosen repeatedly.
02

Applying the Central Limit Theorem

According to the Central Limit Theorem, the mean of the sample means (or the sampling distribution of the sample mean) is equal to the population mean \( \mu \). This is true regardless of the sample size, as long as the sample size is large enough, and the distribution of the sample mean approaches normality.
03

Calculating the Mean of the Sample Means

Since the population mean \( \mu = 495 \), the mean of the average scores (sample means) will also be \( 495 \). This is derived directly from the properties of the Central Limit Theorem.
04

Verifying the Options

The correct option must be \( 495 \) which matches our calculation in Step 3. Options (b) and (c) propose incorrect calculations that do not align with the Central Limit Theorem for this context.

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Key Concepts

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

Normal Distribution
The concept of a normal distribution is fundamental in statistics. A normal distribution is a bell-shaped curve where most of the data points cluster around the mean. In mathematics, it is often represented by the parameters - The mean (average) - The standard deviation (spread of the data)
SAT scores follow this pattern, which helps in making predictions about future outcomes.
A key feature of normal distributions is symmetry about the mean. Fifty percent of the values lie below the mean, and fifty percent lie above. This property allows statisticians to make assumptions about probabilities within that data set based on the distribution's shape.
When you have a large enough sample size, the distribution of many averages will generally approximate a normal distribution, even if the population distribution is not normal. This characteristic is crucial when working with sample means.
Sampling Distribution
A sampling distribution is a distribution of a statistic (like the sample mean) over repeated sampling from a particular population.
For instance, in the SAT problem, a sampling distribution would include the mean scores of many samples of 100 SAT scores. This distribution is instrumental because it helps in understanding how a sample statistic (like a sample mean) relates to the population parameter.
The shape of a sampling distribution tends to be normal if the sample size is large enough, even if the population distribution is not normal.
Furthermore, the mean of the sampling distribution will equal the mean of the population, thanks to the Central Limit Theorem. This concept makes it a powerful tool for inferential statistics allowing us to estimate the likelihood of obtaining certain statistics from random samples.
Standard Deviation
The standard deviation is a measure of how spread out the numbers in a data set are. In the context of SAT scores, the standard deviation was 118, indicating that on average, SAT scores deviate from the mean by 118 points.
A smaller standard deviation means that the data points are closer to the mean, indicating less variability.
In the Central Limit Theorem, the standard deviation of the sampling distribution (sometimes called the standard error) is reduced compared to the population standard deviation and can be calculated as follows: \[ ext{Standard Error} = \frac{\sigma}{\sqrt{n}} \]where \( \sigma \) is the population standard deviation and \( n \) is the sample size. For our example, it's important because it helps us understand the variability in sample means.
Sample Means
Sample means are the averages calculated from samples taken from a population. In our context, these are the average SAT scores from groups of 100 students.
Each time you take a different sample of 100 students, you calculate a new sample mean. When you plot all these means, you get a sampling distribution of the sample mean.
According to the Central Limit Theorem, if you repeat this process many times, the average of all the sample means will equal the population mean, which we deduced as 495 in the SAT problem.
This is significant because it implies that the sample mean is an unbiased estimator of the population mean. In other words, we expect our sample means to center around that population mean when many samples are considered.

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Most popular questions from this chapter

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.

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 283). 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 \(x-\bar{x}\) 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 mean" to see the mean 7 on the graph. Leaving the number of rolls at 1 , click "Roll dice" three times. 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 Figure 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 \(\mathrm{x}^{-} \bar{x}\) 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.

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.

A Sample of Young Men. A government sample survey plans to measure the total cholesterol level of an SRS of men aged \(20-34\). The researchers will report the mean \(x^{-} x\) from their sample as an estimate of the mean total cholesterol level \(\mu\) in this population. (a) Explain to somene who knows no statistics what it means to say that \(x^{-} \bar{x}\) is an "unbiased" estimator of \(\mu\). (b) The sample result \(x^{-} \bar{x}\) is an unbiased estimator of the population truth \(\mu\) no matter what size SRS the study uses. Explain to someone who knows no statistics why a large sample gives more trustworthy results than a small sample.

Measurements in the Lab. Juan makes a measurement in a chemistry laboratory and records the result in his lab report. Suppose that if Juan makes this measurement repeatedly, the standard deviation of his measurements will be \(\sigma=\) 10 milligrams. Juan repeats the measurement four times and records the mean \(x^{-}=\) of his four measurements. (a) What is the standard deviation of Juan's mean result? (That is, if Juan kept on making four measurements and averaging them, what would be the standard deviation of all his \(x^{-} \bar{x}^{\text {'s}} \mathrm{s}\) ?) (b) How many times must Juan repeat the measurement to reduce the standard deviation of \(x^{-} x\) to 2 ? Explain to someone who knows no statistics the advantage of reporting the average of several measurements rather than the result of a single measurement.
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