/*! 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 89 The Critical Reading portion of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 89

The Critical Reading portion of the Scholastic Aptitude Test (SAT) was taken by 1,547,990 college bound students in the class of 2010.31 Scores on that portion of the test range from 200 to 800 and the average score for the class of 2010 is 501 with a standard deviation of 112. Scores are approximately normally distributed. (a) For each sample size below, give the mean and standard deviation of the distribution of the sample means: i. \(n=1\) ii. \(n=10\) iii. \(n=100\) iv. \(n=1000\) (b) Considering your answers from part (a), discuss the effect of the sample size on the center and variability of the distribution of sample means.

Short Answer

Expert verified
a) For all sample sizes of \(n=1, 10, 100, 1000\), the mean of the sample means is 501. The standard deviation of the sample means for \(n=1\) is 112, for \(n=10\) is approximately 35.44, for \(n=100\) is 11.2, and for \(n=1000\) is approximately 3.54. b) As the sample size increases, the standard deviation of the sample means decreases, implying that the larger the sample size, the less variability and the more accurate representation of the population.

Step by step solution

01

Calculate mean of sample means

According to the Central Limit Theorem, the mean of the sample means is equal to the population mean. In this case, the mean score is given as 501. So for all sample sizes \(n=1, 10, 100, 1000\), the mean of the sample means remains 501.
02

Calculate standard deviation of sample means

The standard deviation of the sample means (also known as standard error) can be calculated using the formula \( \sigma / \sqrt{n} \) where \( \sigma \) is the standard deviation of the population and \( n \) is the sample size. Here, the standard deviation of the population is given as 112. Plug these values into the formula to calculate the standard deviation of the sample means for each sample size.
03

Calculate for each sample size

For \(n=1\), the standard deviation of the sample means is \(112 / \sqrt{1} = 112\). For \(n=10\), the standard deviation of the sample means is \(112 / \sqrt{10} \approx 35.44\). For \(n=100\), the standard deviation of the sample means is \(112 / \sqrt{100} = 11.2\). For \(n=1000\), the standard deviation of the sample means is \(112 / \sqrt{1000} \approx 3.54\).
04

Discuss the effect of the sample size on the center and variability

From the calculations, it can be seen that the mean of the sample means (the center of the distribution) remains the same regardless of the sample size. However, the standard deviation of the sample means (which measures variability) decreases as the sample size increases. This is consistent with the concept that increasing the sample size gives a more accurate representation of the population, therefore reducing variability.

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.

Sample Mean
The sample mean is a crucial concept when working with data from a population. It's essentially the average of a set of sample data points. The Central Limit Theorem tells us that regardless of the shape of the population distribution, the distribution of the sample means will tend toward a normal distribution as the sample size increases. This distribution will have a mean equal to the population mean. In the example of SAT scores, given that the average score is 501, any sample mean calculated from random samples of these scores will also average around 501. This remains true irrespective of the sample size.
Standard Deviation
Standard deviation is a measure of variability or the amount of dispersion in a set of values. In the context of sample means, we often refer to the standard error, which is the standard deviation of the sample mean distribution. The formula for the standard error is \( \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) represents the population standard deviation, and \( n \) is the sample size. A smaller standard deviation (or standard error) indicates the sample means are closer together, suggesting less variability. For SAT scores with a population standard deviation of 112, the standard deviation of the sample means decreases as sample size increases, illustrating a reduction in variability as more data points are considered.
Sample Size Effect
The sample size effect is an important consideration when analyzing data. Larger samples provide more information, which typically results in more accurate estimates of the population parameters. The Central Limit Theorem highlights that as the sample size increases, the distribution of sample means will converge more closely to a normal distribution. Additionally, the standard deviation of the sample means will decrease, reflecting less variability and more precision. In the given scenario, increasing the sample size from 1 to 1000 cumulates in a reduction of the sample mean's standard deviation from 112 to approximately 3.54. This illustrates how larger samples lead to a more reliable understanding of the population's characteristics.

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

Refer to a study on hormone replacement therapy. Until 2002 , hormone replacement therapy (HRT), taking hormones to replace those the body no longer makes after menopause, was commonly prescribed to post-menopausal women. However, in 2002 the results of a large clinical trial \(^{56}\) were published, causing most doctors to stop prescribing it and most women to stop using it, impacting the health of millions of women around the world. In the experiment, 8506 women were randomized to take HRT and 8102 were randomized to take a placebo. Table 6.16 shows the observed counts for several conditions over the five years of the study. (Note: The planned duration was 8.5 years. If Exercises 6.205 through 6.208 are done correctly, you will notice that several of the p-values are just below \(0.05 .\) The study was terminated as soon as HRT was shown to significantly increase risk (using a significance level of \(\alpha=0.05)\), because at that point it was unethical to continue forcing women to take HRT). Does HRT influence the chance of a woman getting invasive breast cancer? $$ \begin{array}{lcc} \hline \text { Condition } & \text { HRT Group } & \text { Placebo Group } \\ \hline \text { Cardiovascular Disease } & 164 & 122 \\ \text { Invasive Breast Cancer } & 166 & 124 \\ \text { Cancer (all) } & 502 & 458 \\ \text { Fractures } & 650 & 788 \\ \hline \end{array} $$

Describes scores on the Critical Reading portion of the Scholastic Aptitude Test (SAT) for college-bound students in the class of 2010. Critical Reading scores are approximately normally distributed with mean \(\mu=501\) and standard deviation \(\sigma=112\) (a) For each sample size below, use a normal distribution to find the percentage of sample means that will be greater than or equal to \(525 .\) Assume the samples are random samples. i. \(n=1\) ii. \(n=10\) iii. \(n=100\) iv. \(n=1000\) (b) Considering your answers from part (a), discuss the effect of the sample size on the likelihood of a sample mean being as far from the population mean as \(\bar{x}=525\) is from \(\mu=501\).

Standard Error from a Formula and a Bootstrap Distribution In Exercises 6.19 to \(6.22,\) use StatKey or other technology to generate a bootstrap distribution of sample proportions and find the standard error for that distribution. Compare the result to the standard error given by the Central Limit Theorem, using the sample proportion as an estimate of the population proportion \(p\). Proportion of home team wins in soccer, with \(n=120\) and \(\hat{p}=0.583\)

Situations comparing two proportions are described. In each case, determine whether the situation involves comparing proportions for two groups or comparing two proportions from the same group. State whether the methods of this section apply to the difference in proportions. (a) In a taste test, compare the proportion of tasters who prefer one brand of cola to the proportion who prefer the other brand. (b) Compare the proportion of males who voted in the last election to the proportion of females who voted in the last election. (c) Compare the graduation rate (proportion to graduate) of students on an athletic scholarship to the graduation rate of students who are not on an athletic scholarship. (d) Compare the proportion of voters who vote in favor of a school budget to the proportion who vote against the budget.

According to the 2006 Australia Census, \(^{43} 25.5 \%\) of Australian women over the age of 25 had a college degree, while the percentage for Australian men was \(21.4 \% .\) Suppose we select random samples of 200 women and 200 men from this population and look at the differences in proportions with college degrees, \(\hat{p}_{f}-\hat{p}_{m}\), in those samples. (a) Describe the distribution (center, spread,shape) for the difference in sample proportions. Include a rough sketch of the distribution with values labeled on the horizontal axis. (b) What is the chance that the proportion with college degrees in the men's sample is actually more than the proportion in the women's sample? (Hint: Think about what must be true about \(\hat{p}_{f}-\hat{p}_{m}\) when this happens.)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.