Hypothesis Testing --- One Mean A hypothesis is simply a statement that something is true. Typically, there are two hypotheses in a hypothesis test: the null, and the alternative. Null Hypothesis The hypothesis to be tested is defined as a Null Hypothesis. We use the symbol H o to stand for null hypothesis. Alternative Hypothesis The Alternative Hypothesis is defined as a hypothesis that is considered as an alternative to the null hypothesis. We use the symbol H a to stand for alternative hypothesis. Important Considerations 1. Must contain an equality 2. Tests are always conducted assuming that the null is an equality. 3. The most serious error would be the rejection of a true null. Chapter 9: Hypothesis Testing: One Mean Page -1-
Choosing & Specifying Hypothesis The null hypothesis must always specify a single value when testing population parameters. The alternative hypothesis may be specified as not equal to, greater than, or less than. That is: H o : = 0 H a : 0 --- Two-tailed test H o : = 0 H a : < 0 --- Left-tailed test H o : = 0 H a : > 0 --- Right-tailed test Example: A car manufacturer claims that a certain car gives a gas mileage of 26MPG. Consumer Agency doubts that the claim is true and wants to test it. The hypothesis would be stated as a left-tailed test. The null hypothesis -- H o : µ= 26 The alternate hypothesis -- H a : µ < 26; Chapter 9: Hypothesis Testing: One Mean Page -2-
Pre-Steps To Testing The Hypothesis: 1. Take a random sample of observations gas mileage achieved. 2. Compute the mean. 3. State the hypothesis 4. Test the hypothesis. Motivation For Hypothesis Testing Goal is to test if the population (sample) parameter (statistic) is within a specified acceptance region. If it is then we accept the null hypothesis and conclude that the alternative hypothesis is not true. On the other hand, if the parameter (statistic) value is outside the acceptance region we reject the null hypothesis and conclude that the alternative is true. Some Key Terms 1. The test statistic used to conduct a hypothesis test is: z x n 0 2. Rejection region or Critical region is a set of values for the test statistic that leads to rejection of the null hypothesis. For a two-tailed test the critical region is in both tails. For a left-tailed test the critical region is in the left tail For a right-tailed test the critical region is in the right tail. Chapter 9: Hypothesis Testing: One Mean Page -3-
3. Nonrejection region is a set of values for the test statistic that lead to nonrejection of the null hypothesis. 4. Critical Values are the values of the test statistic that separate the rejection and nonrejection regions. 5. Some Important Critical Values z 0.10 = 1.28 z 0.05 = 1.645 z 0.025 = 1.96 z 0.01 = 2.33 z 0.005 = 2.575 6. Significance level,, of a hypothesis test is defined to be the probability of making a Type I error. Example Cont: Assume = 0.10; critical z = -1.28; n=30; x 25; σ = 1.4; and, n = 0.26 25 26 Then, z 3.91-3.91 < -1.28 thus, reject the null. 0.26 Chapter 9: Hypothesis Testing: One Mean Page -4-
Type I and Type II Errors Decisions\H o True False Do not reject H o Correct Decision Type II Error Reject H o Type I Error Correct Decision Conclusion: Type I error is rejecting the null hypothesis when it is in fact true. Type II error is not rejecting the hypothesis when it is in fact false. The smaller the Type I error probability,, of rejecting a true null hypothesis, the larger the Type II error probability,, of not rejecting a false null hypothesis; and vice versa. Controlling Type I and Type II Errors 1. For a fixed alpha, an increase in sample size will cause a decrease in beta. 2. For any fixed sample size, a decrease in alpha will cause an increase in beta, and vice versa. 3. To decrease both alpha and beta, increase sample size. Example: A person is arrested and we have to decide based on some evidence whether the person is innocent or guilty. Let, Ho = Innocent; and Ha = Guilty. If we convict a person who is innocent we will commit a Type I Error. If a guilty person goes free we have committed a Type II Error. If one chooses a small we will have very few innocent people convicted; but many guilty people might go free. Chapter 9: Hypothesis Testing: One Mean Page -5-
Hypothesis Tests For a Population Mean when Sigma is Known Critical Value Approach 1. State the null and alternative hypothesis 2. Decide on the significance level, 3. Determine the critical values based on the test you are conducting. That is: a) for a two-tailed test CVs are z /2 b) for a left-tailed test CV is -z c) for a right-tailed test CV is z 4. Compute the value of the z test statistic. 5. If the value of the test statistic falls within the rejection region, then reject H o ; otherwise do not reject H o. 6. State the conclusion in words. Possible Conclusion: If the null hypothesis is rejected, we conclude that the alternative hypothesis is probably true. If the null hypothesis is not rejected, we conclude that the data do not provide sufficient evidence to support the alternative hypothesis. See Example 9.5 Chapter 9: Hypothesis Testing: One Mean Page -6-
Example: Cellphone Bill Is there enough evidence to suggest that the mean cellphone bill has decreased from the base year? Given: n=50; = 25; = 0.01 0=47.37; x 41.08 State Hypothesis: Ho: = 47.37 Ha: < 47.37 41.08 47.37 Compute: z 1.78 2.33 25 50 Do not reject the null. We do not have sufficient evidence to conclude the mean bill has decreased from the base year. Chapter 9: Hypothesis Testing: One Mean Page -7-
P-Value Approach The p-value of a hypothesis test is the probability of observing a value of the test statistic that is at least as inconsistent with the null hypothesis as the value of the test statistic actually observed. The p-value is also frequently referred to as the observed significance level or the probability value. Steps: Using the P-Value to Assess the Evidence Against H o Common p-values Evidence against H o P > 0.10 Weak or none 0.05 < P 0.10 Moderate 0.01 < P 0.05 Strong P 0.01 Very strong 1. State the null & alternative hypotheses 2. Decide on the significance level, 3. Compute the value of the test statistic. 4. Determine the p-value for the test statistic. That is, the area in the tail. 5. If the p-value is less than or equal to, reject H o ; otherwise, do not reject H o. 6. State the conclusion in words. Chapter 9: Hypothesis Testing: One Mean Page -8-
P-value for Cellphone Bill Example -- smallest value of alpha to reject the null. = P(z < -1.78) = 0.375 ; since 0.0375 > 0.01 we do not reject the null. Chapter 9: Hypothesis Testing: One Mean Page -9-
Hypothesis Tests For One Population Mean when Sigma is Unknown 1. State the null and alternative hypothesis 2. Decide on the significance level, 3. Determine the critical values based on the test you are conducting with df = n-1. That is: a) for a two-tailed test CVs are t /2 b) for a left-tailed test CV is -t c) for a right-tailed test CV is t 4. Compute the value of the test statistic. NOTE: you will need to compute the standard deviation from the data. 5. If the value of the test statistic falls within the rejection region, then reject H o ; otherwise do not reject H o. 6. State the conclusion in words. See Example: 9.16 Chapter 9: Hypothesis Testing: One Mean Page -10-
Example Bureau of Labor Statistics said that the average consumer unit spent $1749 on apparel and services in 2002. Given that the mean and standard deviation of 25 consumer units in the Northeast is $1935.76 and $350.90 respectively, do the data provide sufficient evidence at the 5% level to conclude that the 2002 mean expenditures in the Northeast are significantly different from the National average? Given: n=25; s = 350.90; = 0.05 0=1749; x 1935.76 State Hypothesis: Ho: = 1749 Ha: <> 1749 1935.76 1749 Compute: t 2.661 2.064 350.90 25 Reject the null hypothesis. We have sufficient evidence to conclude that consumer spending on apparel and service in the Northeast is significantly different from the National average. p-value = 2P(t > 2.661) = 2(0.005) = 0.01 actually it is greater than 0.01 but less than 0.02 Chapter 9: Hypothesis Testing: One Mean Page -11-