/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 Scores on the Evidence-Based Rea... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 15: Problem 20

Scores on the Evidence-Based Reading part of the SAT exam in a recent year were roughly Normal with mean 536 and standard deviation 102. You choose an SRS of 100 students and average their SAT Evidence-Based Reading scores. If you do this many times, the mean of the average scores you get will be close to a. \(536 .\) b. \(536 / 100=5.36\). c. \(536 / \sqrt{100}=53.6\)

Short Answer

Expert verified
The mean of the average scores is 536.

Step by step solution

01

Understand the Problem

The problem asks us to determine the mean of the average scores obtained from a sample of 100 students taken many times from a normally distributed population with a given mean and standard deviation. This relates to the distribution of the sample mean.
02

Recall the Central Limit Theorem

The Central Limit Theorem tells us that the distribution of the sample mean will be normally distributed with the mean equal to the population mean, regardless of the sample size. This implies that the mean of the sample means is the same as the population mean.
03

Identify the Correct Mean

Since the population mean of the SAT Evidence-Based Reading scores is given as 536, and according to the Central Limit Theorem, the mean of the sample means would also be 536 regardless of the sample size of 100 students.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Normal Distribution
A normal distribution, often referred to as a "bell curve" due to its distinct shape, is a fundamental concept in statistics. This distribution is symmetric around its mean, meaning that it has equal tails on both sides. Normal distributions are defined by two key parameters: the mean and the standard deviation. For many natural and standardized processes, such as SAT scores, the data tends to follow a normal distribution. This allows statisticians to make meaningful predictions and calculations.
  • The mean provides the center of the distribution.
  • The standard deviation shows the spread or variability of the scores.
Understanding normal distribution helps us interpret data and its probabilities. It's essential in predicting outcomes based on historical data.
Sample Mean
The sample mean is simply the average of a set of observations from a sample. In the context of the SAT scores problem, if we take several sets of 100 student scores, the sample mean would be the average score within each of these sets. The sample mean is used to estimate the population mean, especially useful when dealing with large data sets where it's impractical to collect every data point.
  • It is calculated by summing all sample data points and dividing by the number of points.
  • The sample mean is a random variable, which will vary depending on the different samples taken.
Notably, the Central Limit Theorem ensures that the distribution of these sample means approaches a normal distribution, making it a reliable approximation of the population mean.
Population Mean
The population mean is the average of all possible observations in a complete population. In our example, the given population mean for the SAT Evidence-Based Reading scores is 536. This is a fixed value representing the average of every student taking the exam that year.
  • It is a constant and doesn鈥檛 change unless the entire population is redefined or altered.
  • The mean serves as a key measure of central tendency, reflecting the center of a distribution.
When using sample means to estimate the population mean, it's crucial to understand that, according to the Central Limit Theorem, the average of these sample means will be approximately equal to the population mean.
Standard Deviation
Standard deviation is a measure of how much the values in a dataset deviate from the mean. It quantifies the amount of variation or dispersion in a set of values. In the SAT score context, a standard deviation of 102 allows us to know how spread out the student scores typically are around the mean score of 536.
  • Standard deviation is symbolized as \( \sigma \) in a population and \( s \) in a sample.
  • A low standard deviation indicates that the scores are closely clustered around the mean, whereas a high standard deviation shows that the scores are more spread out.
This measure is crucial for understanding the reliability and stability of the mean. When sample means are involved, the standard deviation of the sample mean, known as the standard error, is calculated by dividing the population standard deviation by the square root of the sample size, making it valuable in estimating how much sample means vary.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Larger Sample, More Accurate Estimate. Suppose that, in fact, the total cholesterol level of all men aged 20-34 follows the Normal distribution with mean \(\mu=182\) milligrams per deciliter (mg/dL) and standard deviation \(\sigma=37 \mathrm{mg} / \mathrm{dL}\). a. Choose an SRS of 100 men from this population. What is the sampling distribution of \(x\) ? What is the probability that \(x\) takes a value between 180 and 184 \(\mathrm{mg} / \mathrm{dL}\) ? This is the probability that \(x\) estimates \(\mu\) within \(\pm 2 \mathrm{mg} / \mathrm{dL}\). b. Choose an SRS of 1000 men from this population. Now what is the probability that \(x\) falls within \(\pm 2 \mathrm{mg} / \mathrm{dL}\) of \(\mu\) ? The larger sample is much more likely to give an accurate estimate of \(\mu\).

An October 20, 2019, poll of Canadian adults who were registered voters found that \(31.6 \%\) said they would vote conservative in the October 2019 elections. Election records show that \(\mathbf{3 4 . 4} \%\) actually vot ed conservative. The boldface number is a a. sampling distribution. b. statistic. c. parameter.

Sampling Distribution Versus Population Distribution. The 2018 American Time Use Survey contains data on how many minutes of sleep per night each of 9600 survey participants estimated they get. \(-\) The times follow the Normal distribution with mean \(529.2\) minutes and standard deviation \(135.6\) minutes. An SRS of 100 of the participants has a mean time of \(x=514.4\) minutes. A second SRS of size 100 has mean \(x=539.3\) minutes. After many SRSs, the many values of the sample mean \(x\) follow the Normal distribution with mean \(529.9\) minutes and standard deviation \(13.56\) minutes. a. What is the population? What values does the population distribution describe? What is this distribution? b. What values does the sampling distribution of \(x\) describe? What is the sampling distribution?

Pollutants in Aut o Ex hausts (continued). The level of nitrogen oxides (NOX) and nonmethane organic gas (NMOG) in the exhaust over the useful life ( 150,000 miles of driving) of cars of a particular model varies Normally with mean 80 \(\mathrm{mg} / \mathrm{mi}\) and standard deviation \(4 \mathrm{mg} / \mathrm{mi}\). A company has 25 cars of this model in its fleet. What is the level \(L\) such that the probability that the average NOX + NMOG level \(x\) for the fleet is greater than \(L\) is only \(0.01\) ? (Hint: This requires a backward Normal calculation. See page 89 in Chapter 3 if you need a review.)

Testing Glass. How well materials conduct heat matters when designing houses. As a test of a new measurement process, 10 measurements are made on pieces of glass known to have conductivity 1 . The average of the 10 measurements is 1.07. For each of the boldface numbers, indicate whether it is a parameter or a statistic. Explain your answer.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.