Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-1 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-2 Chapter 8 Hypothesis Testing 8-1 Review and Preview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim about a Proportion 8-4 Testing a Claim About a Mean 8-5 Testing a Claim About a Standard Deviation or Variance Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-3 Key Concept This section presents methods for testing a claim about a population mean. Part 1 deals with the very realistic and commonly used case in which the population standard deviation σ is not known. Part 2 discusses the procedure when σ is known, which is very rare. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-4 Part 1 When σ is not known, we use a “t test” that incorporates the Student t distribution. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-5 Notation n = sample size x = sample mean x = population mean Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-6 Requirements 1) The sample is a simple random sample. 2) Either or both of these conditions is satisfied: The population is normally distributed or n > 30. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-7 Test Statistic x x t s n Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-8 Running the Test P-values: Use technology or use the Student t distribution in Table A-3 with degrees of freedom df = n – 1. Critical values: Use the Student t distribution with degrees of freedom df = n – 1. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-9 Important Properties of the Student t Distribution 1. The Student t distribution is different for different sample sizes (see Figure 7-5 in Section 7-3). 2. The Student t distribution has the same general bell shape as the normal distribution; its wider shape reflects the greater variability that is expected when s is used to estimate σ. 3. The Student t distribution has a mean of t = 0. 4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1. 5. As the sample size n gets larger, the Student t distribution gets closer to the standard normal distribution. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-10 Example Listed below are the measured radiation emissions (in W/kg) corresponding to a sample of cell phones. Use a 0.05 level of significance to test the claim that cell phones have a mean radiation level that is less than 1.00 W/kg. 0.38 0.55 1.54 1.55 0.50 The summary statistics are: 0.60 0.92 0.96 1.00 0.86 1.46 x 0.938 and s 0.423 . Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-11 Example - Continued Requirement Check: 1. We assume the sample is a simple random sample. 2. The sample size is n = 11, which is not greater than 30, so we must check a normal quantile plot for normality. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-12 Example - Continued The points are reasonably close to a straight line and there is no other pattern, so we conclude the data appear to be from a normally distributed population. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-13 Example - Continued Step 1: The claim that cell phones have a mean radiation level less than 1.00 W/kg is expressed as μ < 1.00 W/kg. Step 2: The alternative to the original claim is μ ≥ 1.00 W/kg. Step 3: The hypotheses are written as: H 0 : 1.00 W/kg H1 : 1.00 W/kg Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-14 Example - Continued Step 4: The stated level of significance is α = 0.05. Step 5: Because the claim is about a population mean μ, the statistic most relevant to this test is the sample mean: x . Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-15 Example - Continued Step 6: Calculate the test statistic and then find the P-value or the critical value from Table A-3: x x 0.938 1.00 t 0.486 s 0.423 n 11 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-16 Example - Continued Step 7: Critical Value Method: Because the test statistic of t = –0.486 does not fall in the critical region bounded by the critical value of t = –1.812, fail to reject the null hypothesis. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-17 Example - Continued Step 7: P-value method: Technology, such as a TI-83/84 Plus calculator can output the P-value of 0.3191. Since the P-value exceeds α = 0.05, we fail to reject the null hypothesis. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-18 Example Step 8: Because we fail to reject the null hypothesis, we conclude that there is not sufficient evidence to support the claim that cell phones have a mean radiation level that is less than 1.00 W/kg. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-19 Finding P-Values Assuming that neither software nor a TI-83 Plus calculator is available, use Table A-3 to find a range of values for the Pvalue corresponding to the given results. a) In a left-tailed hypothesis test, the sample size is n = 12, and the test statistic is t = –2.007. b) In a right-tailed hypothesis test, the sample size is n = 12, and the test statistic is t = 1.222. c) In a two-tailed hypothesis test, the sample size is n = 12, and the test statistic is t = –3.456. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-20 Example – Confidence Interval Method We can use a confidence interval for testing a claim about μ. For a two-tailed test with a 0.05 significance level, we construct a 95% confidence interval. For a one-tailed test with a 0.05 significance level, we construct a 90% confidence interval. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-21 Example – Confidence Interval Method Using the cell phone example, construct a confidence interval that can be used to test the claim that μ < 1.00 W/kg, assuming a 0.05 significance level. Note that a left-tailed hypothesis test with α = 0.05 corresponds to a 90% confidence interval. Using methods described in Section 7.3, we find: 0.707 W/kg < μ < 1.169 W/kg Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-22 Example – Confidence Interval Method Because the value of μ = 1.00 W/kg is contained in the interval, we cannot reject the null hypothesis that μ = 1.00 W/kg . Based on the sample of 11 values, we do not have sufficient evidence to support the claim that the mean radiation level is less than 1.00 W/kg. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-23 Part 2 When σ is known, we use test that involves the standard normal distribution. In reality, it is very rare to test a claim about an unknown population mean while the population standard deviation is somehow known. The procedure is essentially the same as a t test, with the following exception: Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-24 Test Statistic for Testing a Claim About a Mean (with σ Known) The test statistic is: z x x n The P-value can be provided by technology or the standard normal distribution (Table A-2). The critical values can be found using the standard normal distribution (Table A-2). Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-25 Example If we repeat the cell phone radiation example, with the assumption that σ = 0.480 W/kg, the test statistic is: z x x n 0.938 1.00 0.43 0.480 11 The example refers to a left-tailed test, so the P-value is the area to the left of z = –0.43, which is 0.3336 (found in Table A2). Since the P-value is large, we fail to reject the null and reach the same conclusion as before. Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-26 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Section 8.4-27