FINAL EXAMINATION STUDY MATERIAL III ADDITIONAL READING MATERIAL – INTRO STATS 3RD EDITION CHAPTER 20 PARTS OF CHAPTER 21 PARTS OF CHAPTER 22 PARTS OF CHAPTER 23

Significance Tests About Hypotheses TESTING HYPOTHESES ABOUT CHAPTER 20 Significance Tests About Hypotheses TESTING HYPOTHESES ABOUT PROPORTIONS Copyright © 2013, 2009, and 2007, Pearson Education, Inc.

HYPOTHESIS TESTING ABOUT PROPORTIONS OBJECTIVES DEFINE THE NULL AND ALTERNATIVE HYPOTHESES STATE CONCLUSIONS TO HYPOTHESIS TESTS DISTINGUISH BETWEEN TYPE I AND TYPE II ERRORS

DEFINITION A HYPOTHESIS IS A STATEMENT REGARDING A CHARACTERISTIC OF ONE OR MORE POPULATIONS. HYPOTHESIS TESTING IS A PROCEDURE, BASED ON SAMPLE EVIDENCE AND PROBABILITY, USED TO TEST STATEMENTS REGARDING A CHARACTERISTIC OF ONE OR MORE POPULATION.

PROBLEM FORMULATION SUPPOSE WE TOSSED A COIN 100 TIMES AND WE OBTAINED 38 HEADS AND 62 TAILS. IS THE COIN BIASED? THERE IS NO WAY TO SAY YES OR NO WITH 100% CERTAINTY. BUT WE MAY EVALUATE THE STRENGTH OF SUPPORT TO THE HYPOTHESIS [OR CLAIM] THAT “THE COIN IS BIASED.”

THE NULL AND ALTERNATIVE HYPOTHESES Null hypothesis: Represented by H0, is a statement that there is nothing happening. Generally thought of as the status quo, or no relationship, or no difference. Usually the researcher hopes to disprove or reject the null hypothesis. Alternative hypothesis: Represented by Ha, is a statement that something is happening. In most situations, it is what the researcher hopes to prove. It may be a statement that the assumed status quo is false, or that there is a relationship, or that there is a difference.

NULL AND ALTERNATIVE HYPOTHESES NULL HYPOTHESIS ESTABLISHED FACT; A STATEMENT THAT WE EXPECT DATA TO CONTRADICT; NO CHANGE OF PARAMETERS. ALTERNATIVE HYPOTHESIS NEW CONJECTURE; YOUR CLAIM; A STATEMENT THAT NEEDS A STRONG SUPPORT FROM DATA TO CLAIM IT; CHANGE OF PARAMETERS

NULL AND ALTERNATIVE HYPOTHESES THE NULL HYPOTHESIS IS A STATEMENT TO BE TESTED. IT IS A STATEMENT OF NO CHANGE, NO EFFECT, OR NO DIFFERENCE AND IS ASSUMED TRUE UNTIL EVIDENCE INDICATES OTHERWISE. THE ALTERNATIVE HYPOTHESIS IS A STATEMENT THAT WE ARE TRYING TO FIND EVIDENCE TO SUPPORT.

WRITING NULL AND ALTERNATIVE HYPOTHESES Possible null and alternative hypotheses: 1. H0: p = p0 versus Ha: p  p0 (two-sided) 2. H0: p = p0 versus Ha: p < p0 (one-sided) 3. H0: p = p0 versus Ha: p > p0 (one-sided) p0 = specific value called the null value.

DEMONSTRATIVE EXAMPLE

EXAMPLES WRITE THE NULL AND ALTERNATIVE HYPOTHESES YOU WOULD USE TO TEST EACH OF THE FOLLOWING SITUATIONS. (A) IN THE 1950s ONLY ABOUT 40% OF HIGH SCHOOL GRADUATES WENT ON TO COLLEGE. HAS THE PERCENTAGE CHANGED? (B) 20% OF CARS OF A CERTAIN MODEL HAVE NEEDED COSTLY TRANSMISSION WORK AFTER BEING DRIVEN BETWEEN 50,000 AND 100,000 MILES. THE MANUFACTURER HOPES THAT REDESIGN OF A TRANSMISSION COMPONENT HAS SOLVED THIS PROBLEM. (C) WE FIELD TEST A NEW FLAVOR SOFT DRINK, PLANNING TO MARKET IT ONLY IF WE ARE SURE THAT OVER 60% OF THE PEOPLE LIKE THE FLAVOR.

A significance test about a hypothesis has five steps. The Steps of a Significance Test [Applicable For Proportions and Means] A significance test is a method of using data to summarize the evidence about a hypothesis. A significance test about a hypothesis has five steps. Assumptions Hypotheses Test Statistic P-value Conclusion

Step 1: Assumptions A significance test assumes that the data production used randomization. Other assumptions may include: Assumptions about the sample size [The 10% Condition and Success – Failure Condition]. Assumptions about the shape of the population distribution.

Step 2: Hypothesis A hypothesis is a statement about a population, usually of the form that a certain parameter takes a particular numerical value or falls in a certain range of values. The main goal in many research studies is to check whether the data support certain hypotheses.

Step 2: Hypothesis Each significance test has two hypotheses: The null hypothesis is a statement that the parameter takes a particular value. It has a single parameter value. The alternative hypothesis states that the parameter falls in some alternative range of values.

Null and Alternative Hypotheses The value in the null hypothesis usually represents no effect. The symbol denotes null hypothesis. The value in the alternative hypothesis usually represents an effect of some type. The symbol denotes alternative hypothesis. The alternative hypothesis should express what the researcher hopes to show. The hypotheses should be formulated before viewing or analyzing the data!

Step 3: Test Statistic A test statistic describes how far the point estimate falls from the parameter value given in the null hypothesis (usually in terms of the number of standard errors between the two). If the test statistic falls far from the value suggested by the null hypothesis in the direction specified by the alternative hypothesis, it is evidence against the null hypothesis and in favor of the alternative hypothesis. We use the test statistic to assess the evidence against the null hypothesis by giving a probability, the P-Value.

Step 4: P-value To interpret a test statistic value, we use a probability summary of the evidence against the null hypothesis, . First, we presume that is true. Next, we consider the sampling distribution from which the test statistic comes. We summarize how far out in the tail of this sampling distribution the test statistic falls.

Step 4: P-value We summarize how far out in the tail the test statistic falls by the tail probability of that value and values even more extreme. This probability is called a P-value. The smaller the P-value, the stronger the evidence is against .

Step 4: P-value Figure 12.1 Suppose Were True. The P-value Is the Probability of a Test Statistic Value Like the Observed One or Even More Extreme. This is the shaded area in the tail of the sampling distribution. Question: Which gives stronger evidence against the null hypothesis, a P-value of 0.20 or of 0.01? Why?

Step 4: P-value The P-value is the probability that the test statistic equals the observed value or a value even more extreme. It is calculated by presuming that the null hypothesis is true. The smaller the P-value, the stronger the evidence the data provide against the null hypothesis. That is, a small P-value indicates a small likelihood of observing the sampled results if the null hypothesis were true.

Step 5: Conclusion The conclusion of a significance test reports the P-value and interprets what it says about the question that motivated the test. Sometimes this includes a decision about the validity of the null hypothesis .

BACK TO HYPOTHESIS TESTING ABOUT PROPORTIONS SIGNIFICANCE TESTS BACK TO HYPOTHESIS TESTING ABOUT PROPORTIONS

Steps of a Significance Test about a Population Proportion Step 1: Assumptions The variable is categorical The data are obtained using randomization The 10% Condition The sample size is sufficiently large that the sampling distribution of the sample proportion is approximately normal:

Steps of a Significance Test about a Population Proportion Step 2: Hypotheses The null hypothesis has the form: The alternative hypothesis has the form:

Steps of a Significance Test about a Population Proportion Step 3: Test Statistic The test statistic measures how far the sample proportion falls from the null hypothesis value, , relative to what we’d expect if were true The test statistic is:

Step 3:Test Statistic The z-statistic for the significance test is represents the sample estimate of the proportion p0 represents the specific value in null hypothesis n is the sample size

Steps of a Significance Test about a Population Proportion Step 4: P-value The P-value summarizes the evidence. It describes how unusual the observed data would be if were true.

Computing P - Value For Ha less than, find probability the test statistic z could have been equal to or less than what it is. For Ha greater than, find probability the test statistic z could have been equal to or greater than what it is. For Ha two-sided, p-value includes the probability areas in both extremes of the distribution of the test statistic z.

Computing P - Value

Steps of a Significance Test about a Population Proportion Step 5: Conclusion We summarize the test by reporting and interpreting the P-value.

Using the P – Value to Reach a Conclusion The level of significance, denoted by a (alpha), is a value chosen by the researcher to be the borderline between when a p-value is small enough to choose the alternative hypothesis over the null hypothesis, and when it is not. When the p-value is less than or equal to a, we reject the null hypothesis. When the p-value is larger than a, we cannot reject the null hypothesis. The level of significance may also be called the a-level of the test. Decision: reject H0 if the p-value is smaller than a (usually 0.05, sometimes 0.10 or 0.01). In this case the result is statistically significant

Using the P – Value to Reach a Conclusion When the p-value is small, we reject the null hypothesis or, equivalently, we accept the alternative hypothesis. “Small” is defined as a p- value  a, where a = level of significance (usually 0.05). When the p-value is not small, we conclude that we cannot reject the null hypothesis or, equivalently, there is not enough evidence to reject the null hypothesis. “Not small” is defined as a p-value > a,where a = level of significance (usually 0.05).

Examples From Practice Sheet

How Do We Interpret the P-value? A significance test analyzes the strength of the evidence against the null hypothesis. We start by presuming that is true. The burden of proof is on .

How Do We Interpret the P-value? The approach used in hypothesis testing is called a proof by contradiction. To convince ourselves that is true, we must show that data contradict . If the P-value is small, the data contradict and support .

Two-Sided Significance Tests A two-sided alternative hypothesis has the form The P-value is the two-tail probability under the standard normal curve. We calculate this by finding the tail probability in a single tail and then doubling it.

Summary: P-values for Different Alternative Hypotheses Alternative Hypothesis P-value Right-tail probability Left-tail probability Two-tail probability

Possible Decisions in a Test of Significance Table 12.1 Possible Decisions in a Test of Significance.

“Do Not Reject ” Is Not the Same as Saying “Accept ” Analogy: Legal trial Null Hypothesis: Defendant is Innocent. Alternative Hypothesis: Defendant is Guilty. If the jury acquits the defendant, this does not mean that it accepts the defendant’s claim of innocence. Innocence is plausible, because guilt has not been established beyond a reasonable doubt.

Deciding between a One-Sided and a Two-Sided Test? Things to consider in deciding on the alternative hypothesis: The context of the real problem. In most research articles, significance tests use two-sided P-values. Confidence intervals are two-sided.

TWO SAMPLE POPULATIONS HYPOTHESES TESTING ABOUT DIFFERENCE IN TWO PROPORTIONS

STEPS Step 1 Determine null and alternative hypotheses H0: p1 – p2 = 0 versus Ha: p1 – p2  0 or Ha: p1 – p2 < 0 or Ha: p1 – p2 > 0 Watch how Population 1 and 2 are defined.

STEP 2: Verify Data Condition Samples are independent. Sample sizes are large enough so that – – are at least 5 and preferably at least 10.

Step 2: The Test Statistic Under the null hypothesis, there is a common population proportion p. This common value is estimated using all the data as The Standardized Test Statistics

Step 3: Assume the Null Hypothesis is True, Find the P – value For Ha less than, the p-value is the area below z, even if z is positive. For Ha greater than, the p-value is the area above z, even if z is negative. For Ha two-sided, p-value is 2  area above |z|.

Steps 4 and 5 Decide Whether or Not the Result is Statistically Significant based on p-value and Make a Conclusion in Context Choose a level of significance a, and reject H0 if the p-value is less than (or equal to) a.

Examples From Practice Sheet

Chapter 23 Statistical Inference: Significance Tests About Hypotheses Significance Tests About Means Copyright © 2013, 2009, and 2007, Pearson Education, Inc.

Steps of a Significance Test About a Population Mean Step 1: Assumptions The variable is quantitative. The data are obtained using randomization. The population distribution is approximately normal. This is most crucial when n is small and is one-sided.

Steps of a Significance Test About a Population Mean Step 2: Hypotheses: The null hypothesis has the form: The alternative hypothesis has the form:

Steps of a Significance Test About a Population Mean Step 3: Test Statistic The test statistic measures how far the sample mean falls from the null hypothesis value , as measured by the number of standard errors between them. The test statistic is:

Steps of a Significance Test About a Population Mean Step 4: P-value The P-value summarizes the evidence. It describes how unusual the data would be if were true.

Summary of P-values for Different Alternative Hypotheses The P-value is a single tail or a two-tail probability depending on whether the alternative hypothesis is one-sided or two-sided. Alternative Hypothesis P-value Right-tail probability from t distribution Left-tail probability from t distribution Two-tail probability from t distribution

Steps of a Significance Test About a Population Mean Step 5: Conclusion We summarize the test by reporting and interpreting the P-value.

Results of Two-Sided Tests and Results of Confidence Intervals Agree Conclusions about means using two-sided significance tests are consistent with conclusions using confidence intervals. If P-value in a two-sided test, a 95% confidence interval does not contain the value specified by the null hypothesis. If P-value in a two-sided test, a 95% confidence interval does contain the value specified by the null hypothesis.

Examples From Practice Sheet

When the Population Does Not Satisfy the Normality Assumption For large samples (roughly about 30 or higher), this assumption is usually not important. The sampling distribution of is approximately normal regardless of the population distribution.

When the Population Does Not Satisfy the Normality Assumption In the case of small samples, we cannot assume that the sampling distribution of is approximately normal. Two-sided inferences using the t distribution are robust against violations of the normal population assumption. They still usually work well if the actual population distribution is not normal. The test does not work well for a one-sided test with small n when the population distribution is highly skewed.

Regardless of Robustness, Look at the Data Whether n is small or large, you should look at the data to check for severe skew or for outliers that occur primarily in one direction. They could cause the sample mean to be a misleading measure.

Chapter 21 Statistical Inference: Significance Tests About Hypotheses Decisions and Types of Errors in Significance Tests Copyright © 2013, 2009, and 2007, Pearson Education, Inc.

Type I and Type II Errors When is true, a Type I Error occurs when is rejected. When is false, a Type II Error occurs when is not rejected. As P(Type I Error) goes Down, P(Type II Error) goes Up. The two probabilities are inversely related.

Significance Test Results Table 9.6 The Four Possible Results of a Decision in a Significance Test. Type I and Type II errors are the two possible incorrect decisions. We make a correct decision if we do not reject when it is true or if we reject it when it is false.

Table 9.7 Possible Results of a Legal Trial Significance Test Results Table 9.7 Possible Results of a Legal Trial

Decision Errors: Type I If we reject when in fact is true, this is a Type I error. If we decide there is a significant relationship in the population (reject the null hypothesis): This is an incorrect decision only if is true. The probability of this incorrect decision is equal to . If we reject the null hypothesis when it is true and : There really is no relationship and the extremity of the test statistic is due to chance. About 5% of all samples from this population will lead us to incorrectly reject the null hypothesis and conclude significance. 65

P(Type I Error) = Significance Level Suppose is true. The probability of rejecting , thereby committing a Type I error, equals the significance level, , for the test. We can control the probability of a Type I error by our choice of the significance level. The more serious the consequences of a Type I error, the smaller should be. For example, suppose a convicted defendant gets the death penalty. Then, if a defendant is actually innocent, we would hope that the probability of conviction is smaller than 0.05.

Decision Errors: Type II If we fail to reject when in fact is true, this is a Type II error. If we decide not to reject the null hypothesis and thus allow for the plausibility of the null hypothesis. We make an incorrect decision only if is true. The probability of this incorrect decision is denoted by . We’ll learn how to calculate P(Type II error) later in the chapter. 67

Power of a Test When is false, you want the probability of rejecting it to be high. The probability of rejecting when it is false is called the power of the test. Power = 1 – P(Type II error) for a particular value of the parameter from the range of alternative hypothesis values. The higher the power, the better. In practice, it is ideal for studies to have high power while using a relatively small significance level.

Examples From Practice Sheet