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AP Stats 91 Slides with annotations.notebook
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November 17, 2011
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The Practice of Statistics, 4th edition – For AP*STARNES, YATES, MOORE
Chapter 9: Testing a ClaimSection 9.1Significance Tests: The Basics
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Chapter 9Testing a Claim
• 9.1 Significance Tests: The Basics• 9.2 Tests about a Population Proportion• 9.3 Tests about a Population Mean
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Section 9.1Significance Tests: The Basics
After this section, you should be able to…• STATE correct hypotheses for a significance test about a population proportion or mean.• INTERPRET Pvalues in context.• INTERPRET a Type I error and a Type II error in context, and give the consequences of each.• DESCRIBE the relationship between the significance level of a test, P(Type II error), and power.
Learning Objectives
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Significance Tests: The Basics
• IntroductionConfidence intervals are one of the two most common types of statistical inference. Use a confidence interval when your goal is to estimate a population parameter. The second common type of inference, called significance tests, has a different goal: to assess the evidence provided by data about some claim concerning a population.A significance test is a formal procedure for comparing observed data with a claim (also called a hypothesis) whose truth we want to assess. The claim is a statement about a parameter, like the population proportion p or the population mean µ. We express the results of a significance test in terms of a probability that measures how well the data and the claim agree.In this chapter, we’ll learn the underlying logic of statistical tests, how to perform tests about population proportions and population means, and how tests are connected to confidence intervals.
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http://bcs.whfreeman.com/tps4e/#628644__666398__
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AP Stats 91 Slides with annotations.notebook
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The claim (hypothesis) is a statement about a population parameter! Therefore we should use the appropriate symbols p and μ when stating hypotheses.
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• Stating HypothesesA significance test starts with a careful statement of the claims we want to compare. The first claim is called the null hypothesis. Usually, the null hypothesis is a statement of “no difference.” The claim we hope or suspect to be true instead of the null hypothesis is called the alternative hypothesis.
Significance Tests: The Basics
Definition:The claim tested by a statistical test is called the null hypothesis (H0). The test is designed to assess the strength of the evidence against the null hypothesis. Often the null hypothesis is a statement of “no difference.” The claim about the population that we are trying to find evidence for is the alternative hypothesis (Ha).
In the freethrow shooter example, our hypotheses are H0 : p = 0.80Ha : p < 0.80where p is the longrun proportion of made free throws.
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Example
Don’t argue! A Gallup Poll report on a national survey of 1028 teenagers revealed that 72% of teens said they seldom or never argue with their friends.7 Yvonne wonders whether this national result would be true in her large high school. So she surveys a random sample of 150 students at her school.
State the appropriate null hypothesis and alternative hypothesis in each case. Be sure to define your parameter each time.
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AP Stats 91 Slides with annotations.notebook
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What is wrong with the stated hypotheses? How should it be fixed?
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Don't Argue Example Again
For Yvonne’s survey, 96 students in the sample said they rarely or never argue with friends. A significance test yields a Pvalue of 0.0291.(a) Interpret this result in context.(b) Do the data provide convincing evidence against the null hypothesis? Explain.
Don’t argue! A Gallup Poll report on a national survey of 1028 teenagers revealed that 72% of teens said they seldom or never argue with their friends.7 Yvonne wonders whether this national result would be true in her large high school. So she surveys a random sample of 150 students at her school.
If the proportion of students at Yvonne's school who say they rarely or never argue with friends is really 0.72, there is a 2.91% chance of finding a sample of 150 students with a value of p-hat that is as far from 0.72 as the sample value in either direction.
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Criminal CasesA person is innocent until proven otherwise.
Type I Error:
Type II Error:
Which is worse in this scenario?
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Section 9.1Significance Tests: The Basics
In this section, we learned that…• A significance test assesses the evidence provided by data against a null hypothesis H0 in favor of an alternative hypothesis Ha.• The hypotheses are stated in terms of population parameters. Often, H0 is a statement of no change or no difference. Ha says that a parameter differs from its null hypothesis value in a specific direction (onesided alternative) or in either direction (twosided alternative).• The reasoning of a significance test is as follows. Suppose that the null hypothesis is true. If we repeated our data production many times, would we often get data as inconsistent with H0 as the data we actually have? If the data are unlikely when H0 is true, they provide evidence against H0 .• The Pvalue of a test is the probability, computed supposing H0 to be true, that the statistic will take a value at least as extreme as that actually observed in the direction specified by Ha .
Summary
AP Stats 91 Slides with annotations.notebook
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November 17, 2011
Nov 1712:45 PM
EXAM WILL BE ON WEDNESDAY NEXT WEEK.
IT IS A SHORTENED CLASS PERIOD, AND THEREFORE WILL NOT BE AS LONG OF AN EXAM. IT WILL INCLUDE BOTH CH 8 AND CH 9 MATERIAL.
Nov 1712:48 PM
On the waxed paper provided, answer the following questions please and number them.
1) Completed all Ch. 8 Guided Notes2) Completed all Ch. 8 Assignments1) Completed 9.1 Guided Notes2) Read 9.1 Prior to Class
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Section 9.1Significance Tests: The Basics
• Small Pvalues indicate strong evidence against H0 . To calculate a Pvalue, we must know the sampling distribution of the test statistic when H0 is true. There is no universal rule for how small a Pvalue in a significance test provides convincing evidence against the null hypothesis.• If the Pvalue is smaller than a specified value α (called the significance level), the data are statistically significant at level α. In that case, we can reject H0 . If the Pvalue is greater than or equal to α, we fail to reject H0 .• A Type I error occurs if we reject H0 when it is in fact true. A Type II error occurs if we fail to reject H0 when it is actually false. In a fixed level α significance test, the probability of a Type I error is the significance level α.• The power of a significance test against a specific alternative is the probability that the test will reject H0 when the alternative is true. Power measures the ability of the test to detect an alternative value of the parameter. For a specific alternative, P(Type II error) = 1 power.
Summary
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