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Chapter 9 Hypothesis Testing McGraw-Hill/Irwin Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Hypothesis Testing 9.1 9.2 9.3 9.4 9.5 9.6 9.7 Null and Alternative Hypotheses and Errors in Testing z Tests about a Population Mean s Known t Tests about a Population Mean s Unknown z Tests about a Population Proportion Type II Error Probabilities and Sample Size Determination (Optional) The Chi-Square Distribution (Optional) Statistical Inference for a Population Variance (Optional) 9-2 LO 1: Specify appropriate null and alternative hypotheses. 9.1 Null and Alternative Hypotheses and Errors in Hypothesis Testing Null hypothesis, H0, is a statement of the basic proposition being tested Represents the status quo and is not rejected unless there is convincing sample evidence that it is false Alternative hypothesis, Ha, is an alternative accepted only if there is convincing sample evidence it is true One-Sided, “Greater Than” H0: 0 vs. Ha: > 0 One-Sided, “Less Than” H0 : 0 vs. Ha : < 0 Two-Sided, “Not Equal To” H0 : = 0 vs. Ha : 0 where 0 is a given constant value (with the appropriate units) that is a comparative value 9-3 LO 2: Describe Type I and Type II errors and their probabilities. Error Probabilities Type I Error: Rejecting H0 when it is true is the probability of making a Type I error 1 – is the probability of not making a Type I error Type II Error: Failing to reject H0 when it is false β is the probability of making a Type II error 1 – β is the probability of not making a Type II error State of Nature Conclusion Reject H0 Do not Reject H0 H0 True H0 False Type I Error (α Correct Decision Correct Decision Type II Error (β ) ) 9-4 LO 3: Use critical values and p-values to perform a z test about a population mean when s is known. 9.2 z Tests about a Population Mean: σ Known Test hypotheses about a population mean using the normal distribution Called z tests Require that the true value of the population standard deviation σ is known In most real-world situations, σ is not known But often is estimated from s of a single sample When σ is unknown, test hypotheses about a population mean using the t distribution Here, assume that we know σ 9-5 LO3 1. 2. 3. 4. 5. 6. 7. Steps in Testing a “Greater Than” Alternative State the null and alternative hypotheses Specify the significance level Select the test statistic Determine the critical value rule for deciding whether or not to reject H0 Collect the sample data and calculate the value of the test statistic Decide whether to reject H0 by using the test statistic and the rejection rule Interpret the statistical results in managerial terms and assess their practical importance 9-6 LO 4: Use critical values and p-values to perform a t test about a population mean. Assume the population being sampled is normally distributed The population standard deviation σ is unknown, as is the usual situation 9.3 t Tests about a Population Mean: σ Unknown If the population standard deviation σ is unknown, then it will have to estimated from a sample standard deviation s Under these two conditions, have to use the t distribution to test hypotheses 9-7 LO4 Defining the t Statistic: σ Unknown Let x be the mean of a sample of size n with standard deviation s Also, µ0 is the claimed value of the population mean Define a new test statistic x 0 t s n If the population being sampled is normal, and s is used to estimate σ, then … The sampling distribution of the t statistic is a t distribution with n – 1 degrees of freedom 9-8 LO4 t Tests about a Population Mean: σ Unknown Alternative Reject H0 if: p-value Ha: µ > µ0 t > t Area under t distribution to right of t Ha: µ < µ0 t < –t Area under t distribution to left of –t Ha: µ µ0 |t| > t /2 * Twice area under t distribution to right of |t| tα, tα/2, and p-values are based on n – 1 degrees of freedom (for a sample of size n) * either t > tα/2 or t < –tα/2 9-9 LO 5: Use critical values and p-values to perform a large sample z test about a population proportion. 9.4 z Tests about a Population Proportion Alternative Reject H0 if: p-value Ha: ρ > ρ0 z > z Area under t distribution to right of z Ha: ρ < ρ0 z < –z Area under t distribution to left of –z Ha: ρ ρ0 |z| > z /2 * Twice area under t distribution to right of |z| Where the test statistics is * either z > zα/2 or z < –zα/2 z p̂ p0 p0 1 p0 n 9-10 LO 6: Calculate Type II error probabilities and the power of a test, and determine sample size (optional). Want the probability β of not rejecting a false null hypothesis That is, want the probability β of committing a Type II error 1 - β is called the power of the test Assume that the sampled population is normally distributed, or that a large sample is taken Test… 9.5 Type II Error Probabilities and Sample Size Determination (Optional) H0: µ = µ0 vs Ha: µ < µ0 or Ha: µ > µ0 or Ha: µ ≠ µ0 Want to make the probability of a Type I error equal to α and randomly select a sample of size n 9-11 LO6 Calculating β The probability β of a Type II error corresponding to the alternative value µa for µ is equal to the area under the standard normal curve to the left of z* Continued 0 a s n Here z* equals zα if the alternative hypothesis is one-sided (µ < µ0 or µ > µ0) Also z* ≠ zα/2 if the alternative hypothesis is twosided (µ ≠ µ0) 9-12 LO6 Sample Size Population is normal, or a large sample is taken H0: = 0 vs. Ha: < 0 or Ha: > 0 or Ha: ≠ 0 Want the probability of a Type I error equal to and probability of a Type II error corresponding to alternative value for equal to b Then take a sample of size: z * zb 2 s 2 n 0 a 2 z* equals z, if alternative hypothesis is one-sided and z* equals z/2 if the alternative hypothesis is two-sided zb is point on the standard normal curve that gives a right-hand tail area equal to b 9-13 LO 7: Describe the properties of the chisquare distribution and use a chi-square table (optional). The chi-square ² distribution depends on the number of degrees of freedom 9.6 The Chi-Square Distribution (Optional) See Table A.17 of Appendix A A chi-square point ²α is the point under a chisquare distribution that gives right-hand tail area 9-14 LO 8: Use the chisquare distribution to make statistical inferences about population variances (optional). 9.7 Statistical Inference for Population Variance (Optional) If s2 is the variance of a random sample of n measurements from a normal population with variance σ2 The sampling distribution of the statistic (n - 1) s2 / σ2 is a chi-square distribution with (n – 1) degrees of freedom Can calculate confidence interval and perform hypothesis testing 100(1-α)% confidence interval for σ2 (n 1) s 2 (n 1) s 2 , 2 2 1 / 2 / 2 9-15