Book cover for Stats Data and Models

Stats Data and Models

Richard D. De Veaux, Paul F. Velleman, David E. Bock

ISBN #9780321986498

4th Edition

1,406 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes the use of Student's t-distributions for inference about means when the population standard deviation is unknown. By replacing ? with the sample standard deviation s, we accommodate extra variability through wider confidence intervals and adjusted P-values, especially for small sample sizes. A proper t-interval and t-test require verifying assumptions such as independence, randomness, and nearly Normal distribution of the data. Moreover, the methodology extends to determining sample sizes needed for a desired margin of error, ensuring that inferences made about the population mean are both accurate and reliable.

Learning Objectives

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Key Concepts

CONCEPT

DEFINITION

Analysis of Variance (ANOVA)

A collection of statistical techniques used to compare the means of three or more groups by partitioning total variability into components due to differences among groups (between-group) and within groups.

Example Problems

Example 1

A specialty food company sells whole King Salmon to various customers. The mean weight of these salmon is 37 pounds with a standard deviation of 4 pounds. The company ships them to restaurants in boxes of 4 salmon, to grocery stores in cartons of 16 salmon, and to discount outlet stores in pallets of 100 salmon. To forecast costs, the shipping department needs to estimate the standard deviation of the mean weight of the salmon in each type of shipment a) Find the standard deviations of the mean weight of the salmon in each type of shipment. b) The distribution of the salmon weights turns out to be skewed to the high end. Would the distribution of shipping weights be better characterized by a Normal model for the boxes or pallets? Explain.

Example 2

The LSAT (a test taken for law school admission) has a mean score of 151 with a standard deviation of 9 and a unimodal, symmetric distribution of scores. A test preparation organization teaches small classes of 9 students at a time. A larger organization teaches classes of 25 students at a time. Both organizations publish the mean scores of all their classes. a) What would you expect the distribution of mean class scores to be for each organization? b) If either organization has a graduating class with a mean score of $160,$ they'll take out a full-page ad in the local school paper to advertise. Which organization is more likely to have that success? Explain. c) Both organizations advertise that if any class has an average score below $145,$ they'll pay for everyone to retake the LSAT. Which organization is at greater risk to have to pay?

Example 3

$t$ -models, part I Using the $t$ tables, software, or a calculator, estimate a) the critical value of $t$ for a $99 \%$ confidence interval with $\mathrm{df}=12$. b) the critical value of $t$ for a $98 \%$ confidence interval with $\mathrm{df}=92$.

Example 4

$t$ -models, part II Using the $t$ tables, software, or a calculator, estimate a) the critical value of $t$ for a $99 \%$ confidence interval with $\mathrm{df}=2$ b) the critical value of $t$ for a $90 \%$ confidence interval with $\mathrm{df}=111$

Example 5

Describe how the shape, center, and spread of $t$ -models change as the number of degrees of freedom increases.
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Step-by-Step Explanations

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Common Mistakes

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