© 2010 Pearson Prentice Hall. All rights reserved Chapter Hypothesis Tests Regarding a Parameter 10

© 2010 Pearson Prentice Hall. All rights reserved Section Hypothesis Tests for a Population Mean-Population Standard Deviation Known 10.2

© 2010 Pearson Prentice Hall. All rights reserved Objectives 1.Explain the logic of hypothesis testing 2.Test the hypotheses about a population mean with  known using the classical approach 3.Test hypotheses about a population mean with  known using P-values 4.Test hypotheses about a population mean with  known using confidence intervals 5.Distinguish between statistical significance and practical significance.

© 2010 Pearson Prentice Hall. All rights reserved Objective 1 Explain the Logic of Hypothesis Testing

© 2010 Pearson Prentice Hall. All rights reserved To test hypotheses regarding the population mean assuming the population standard deviation is known, two requirements must be satisfied: 1.A simple random sample is obtained. 2.The population from which the sample is drawn is normally distributed or the sample size is large (n≥30). If these requirements are met, the distribution of is normal with mean  and standard deviation.

© 2010 Pearson Prentice Hall. All rights reserved Recall the researcher who believes that the mean length of a cell phone call has increased from its March, 2006 mean of 3.25 minutes. Suppose we take a simple random sample of 36 cell phone calls. Assume the standard deviation of the phone call lengths is known to be 0.78 minutes. What is the sampling distribution of the sample mean? Answer: is normally distributed with mean 3.25 and standard deviation.

© 2010 Pearson Prentice Hall. All rights reserved Suppose the sample of 36 calls resulted in a sample mean of 3.56 minutes. Do the results of this sample suggest that the researcher is correct? In other words, would it be unusual to obtain a sample mean of 3.56 minutes from a population whose mean is 3.25 minutes? What is convincing or statistically significant evidence?

© 2010 Pearson Prentice Hall. All rights reserved When observed results are unlikely under the assumption that the null hypothesis is true, we say the result is statistically significant. When results are found to be statistically significant, we reject the null hypothesis.

© 2010 Pearson Prentice Hall. All rights reserved One criterion we may use for sufficient evidence for rejecting the null hypothesis is if the sample mean is too many standard deviations from the assumed (or status quo) population mean. For example, we may choose to reject the null hypothesis if our sample mean is more than 2 standard deviations above the population mean of 3.25 minutes. The Logic of the Classical Approach

© 2010 Pearson Prentice Hall. All rights reserved Recall that our simple random sample of 36 calls resulted in a sample mean of 3.56 minutes with standard deviation of Thus, the sample mean is standard deviations above the hypothesized mean of 3.25 minutes. Therefore, using our criterion, we would reject the null hypothesis and conclude that the mean cellular call length is greater than 3.25 minutes.

© 2010 Pearson Prentice Hall. All rights reserved Why does it make sense to reject the null hypothesis if the sample mean is more than 2 standard deviations above the hypothesized mean?

© 2010 Pearson Prentice Hall. All rights reserved If the null hypothesis were true, then =0.9772=97.72% of all sample means will be less than (0.13)=3.51.

© 2010 Pearson Prentice Hall. All rights reserved Because sample means greater than 3.51 are unusual if the population mean is 3.25, we are inclined to believe the population mean is greater than 3.25.

© 2010 Pearson Prentice Hall. All rights reserved A second criterion we may use for sufficient evidence to support the alternative hypothesis is to compute how likely it is to obtain a sample mean at least as extreme as that observed from a population whose mean is equal to the value assumed by the null hypothesis. The Logic of the P-Value Approach

© 2010 Pearson Prentice Hall. All rights reserved We can compute the probability of obtaining a sample mean of 3.56 or more using the normal model.

© 2010 Pearson Prentice Hall. All rights reserved Recall So, we compute The probability of obtaining a sample mean of 3.56 minutes or more from a population whose mean is 3.25 minutes is This means that fewer than 1 sample in 100 will give us a mean as high or higher than 3.56 if the population mean really is 3.25 minutes. Since this outcome is so unusual, we take this as evidence against the null hypothesis.

© 2010 Pearson Prentice Hall. All rights reserved Assuming that H 0 is true, if the probability of getting a sample mean as extreme or more extreme than the one obtained is small, we reject the null hypothesis. Premise of Testing a Hypothesis Using the P-value Approach

© 2010 Pearson Prentice Hall. All rights reserved Objective 2 Test Hypotheses about a Population Mean with  Known Using the Classical Approach

© 2010 Pearson Prentice Hall. All rights reserved Testing Hypotheses Regarding the Population Mean with σ Known Using the Classical Approach To test hypotheses regarding the population mean with  known, we can use the steps that follow, provided that two requirements are satisfied: 1.The sample is obtained using simple random sampling. 2.The sample has no outliers, and the population from which the sample is drawn is normally distributed or the sample size is large (n ≥ 30).

© 2010 Pearson Prentice Hall. All rights reserved Step 1: Determine the null and alternative hypotheses. Again, the hypotheses can be structured in one of three ways:

© 2010 Pearson Prentice Hall. All rights reserved Step 2: Select a level of significance, , based on the seriousness of making a Type I error.

© 2010 Pearson Prentice Hall. All rights reserved Step 3: Provided that the population from which the sample is drawn is normal or the sample size is large, and the population standard deviation, , is known, the distribution of the sample mean,, is normal with mean and standard deviation. Therefore, represents the number of standard deviations that the sample mean is from the assumed mean. This value is called the test statistic.

© 2010 Pearson Prentice Hall. All rights reserved Step 4: The level of significance is used to determine the critical value. The critical region represents the maximum number of standard deviations that the sample mean can be from  0 before the null hypothesis is rejected. The critical region or rejection region is the set of all values such that the null hypothesis is rejected.

© 2010 Pearson Prentice Hall. All rights reserved (critical value) Two-Tailed

© 2010 Pearson Prentice Hall. All rights reserved (critical value) Left-Tailed

© 2010 Pearson Prentice Hall. All rights reserved Right-Tailed (critical value)

© 2010 Pearson Prentice Hall. All rights reserved Step 5: Compare the critical value with the test statistic:

© 2010 Pearson Prentice Hall. All rights reserved Step 6: State the conclusion.

© 2010 Pearson Prentice Hall. All rights reserved The procedure is robust, which means that minor departures from normality will not adversely affect the results of the test. However, for small samples, if the data have outliers, the procedure should not be used.

© 2010 Pearson Prentice Hall. All rights reserved Parallel Example 2: The Classical Approach to Hypothesis Testing A can of 7-Up states that the contents of the can are 355 ml. A quality control engineer is worried that the filling machine is miscalibrated. In other words, she wants to make sure the machine is not under- or over-filling the cans. She randomly selects 9 cans of 7-Up and measures the contents. She obtains the following data Is there evidence at the  =0.05 level of significance to support the quality control engineer’s claim? Prior experience indicates that  =3.2ml. Source: Michael McCraith, Joliet Junior College

© 2010 Pearson Prentice Hall. All rights reserved Solution The quality control engineer wants to know if the mean content is different from 355 ml. Since the sample size is small, we must verify that the data come from a population that is approximately normal with no outliers.

© 2010 Pearson Prentice Hall. All rights reserved Assumption of normality appears reasonable. Normal Probability Plot for Contents (ml)

© 2010 Pearson Prentice Hall. All rights reserved No outliers.

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 1: H 0 :  =355 versus H 1 :  ≠355 Step 2: The level of significance is  =0.05. Step 3: The sample mean is calculated to be The test statistic is then The sample mean of is 1.56 standard deviations above the assumed mean of 355 ml.

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 4: Since this is a two-tailed test, we determine the critical values at the  =0.05 level of significance to be -z = and z =1.96 Step 5: Since the test statistic, z 0 =1.56, is less than the critical value 1.96, we fail to reject the null hypothesis. Step 6: There is insufficient evidence at the  =0.05 level of significance to conclude that the mean content differs from 355 ml.

© 2010 Pearson Prentice Hall. All rights reserved Objective 3 Test Hypotheses about a Population Mean with  Known Using P-values.

© 2010 Pearson Prentice Hall. All rights reserved A P-value is the probability of observing a sample statistic as extreme or more extreme than the one observed under the assumption that the null hypothesis is true.

© 2010 Pearson Prentice Hall. All rights reserved Testing Hypotheses Regarding the Population Mean with σ Known Using P-values To test hypotheses regarding the population mean with  known, we can use the steps that follow to compute the P-value, provided that two requirements are satisfied: 1.The sample is obtained using simple random sampling. 2.The sample has no outliers, and the population from which the sample is drawn is normally distributed or the sample size is large (n ≥ 30).

© 2010 Pearson Prentice Hall. All rights reserved Step 1: A claim is made regarding the population mean. The claim is used to determine the null and alternative hypotheses. Again, the hypothesis can be structured in one of three ways:

© 2010 Pearson Prentice Hall. All rights reserved Step 2: Select a level of significance, , based on the seriousness of making a Type I error.

© 2010 Pearson Prentice Hall. All rights reserved Step 3: Compute the test statistic,

© 2010 Pearson Prentice Hall. All rights reserved Step 4: Determine the P-value

© 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved

© 2010 Pearson Prentice Hall. All rights reserved Step 5: Reject the null hypothesis if the P-value is less than the level of significance, . The comparison of the P-value and the level of significance is called the decision rule. Step 6: State the conclusion.

© 2010 Pearson Prentice Hall. All rights reserved Parallel Example 3: The P-Value Approach to Hypothesis Testing: Left-Tailed, Large Sample The volume of a stock is the number of shares traded in the stock in a day. The mean volume of Apple stock in 2007 was million shares with a standard deviation of million shares. A stock analyst believes that the volume of Apple stock has increased since then. He randomly selects 40 trading days in 2008 and determines the sample mean volume to be million shares. Test the analyst’s claim at the  =0.10 level of significance using P-values.

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 1: The analyst wants to know if the stock volume has increased. This is a right-tailed test with H 0 :  =35.14 versus H 1 :  > We want to know the probability of obtaining a sample mean of or more from a population where the mean is assumed to be

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 2: The level of significance is  =0.10. Step 3: The test statistic is

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 4: P(Z > z 0 )=P(Z > 2.48)= The probability of obtaining a sample mean of or more from a population whose mean is is

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 5: Since the P-value= is less than the level of significance, 0.10, we reject the null hypothesis. Step 6: There is sufficient evidence to reject the null hypothesis and to conclude that the mean volume of Apple stock is greater than million shares.

© 2010 Pearson Prentice Hall. All rights reserved Parallel Example 4: The P-Value Approach of Hypothesis Testing: Two-Tailed, Small Sample A can of 7-Up states that the contents of the can are 355 ml. A quality control engineer is worried that the filling machine is miscalibrated. In other words, she wants to make sure the machine is not under- or over-filling the cans. She randomly selects 9 cans of 7-Up and measures the contents. She obtains the following data Use the P-value approach to determine if there is evidence at the  =0.05 level of significance to support the quality control engineer’s claim. Prior experience indicates that  =3.2ml. Source: Michael McCraith, Joliet Junior College

© 2010 Pearson Prentice Hall. All rights reserved Solution The quality control engineer wants to know if the mean content is different from 355 ml. Since we have already verified that the data come from a population that is approximately normal with no outliers, we will continue with step 1.

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 1: The quality control engineer wants to know if the content has changed. This is a two-tailed test with H 0 :  =355 versus H 1 :  ≠355.

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 2: The level of significance is  =0.05. Step 3: Recall that the sample mean is The test statistic is then

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 4: Since this is a two-tailed test, P-value = P(Z 1.56) = 2*(0.0594)= The probability of obtaining a sample mean that is more than 1.56 standard deviations away from the assumed mean of 355 ml is

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 5: Since the P-value= is greater than the level of significance, 0.05, we fail to reject the null hypothesis. Step 6: There is insufficient evidence to conclude that the mean content of 7-Up cans differs from 355 ml.

© 2010 Pearson Prentice Hall. All rights reserved One advantage of using P-values over the classical approach in hypothesis testing is that P-values provide information regarding the strength of the evidence. Another is that P- values are interpreted the same way regardless of the type of hypothesis test being performed. the lower the P-value, the stronger the evidence against the statement in the null hypothesis.

© 2010 Pearson Prentice Hall. All rights reserved Objective 4 Test Hypotheses about a Population Mean with  Known Using Confidence Intervals

© 2010 Pearson Prentice Hall. All rights reserved When testing H 0 :  =  0 versus H 1 :  ≠  0, if a (1-  )·100% confidence interval contains  0, we do not reject the null hypothesis. However, if the confidence interval does not contain  0, we have sufficient evidence that supports the statement in the alternative hypothesis and conclude that  ≠  0 at the level of significance, .

© 2010 Pearson Prentice Hall. All rights reserved Parallel Example 6: Testing Hypotheses about a Population Mean Using a Confidence Interval Test the hypotheses presented in Parallel Examples 2 and 4 at the  =0.05 level of significance by constructing a 95% confidence interval about , the population mean can content.

© 2010 Pearson Prentice Hall. All rights reserved Solution Lower bound: Upper bound: We are 95% confident that the mean can content is between ml and ml. Since the mean stated in the null hypothesis is in this interval, there is insufficient evidence to reject the hypothesis that the mean can content is 355 ml.

© 2010 Pearson Prentice Hall. All rights reserved Objective 5 Distinguish between Statistical Significance and Practical Significance

© 2010 Pearson Prentice Hall. All rights reserved When a large sample size is used in a hypothesis test, the results could be statistically significant even though the difference between the sample statistic and mean stated in the null hypothesis may have no practical significance.

© 2010 Pearson Prentice Hall. All rights reserved Practical significance refers to the idea that, while small differences between the statistic and parameter stated in the null hypothesis are statistically significant, the difference may not be large enough to cause concern or be considered important.

© 2010 Pearson Prentice Hall. All rights reserved Parallel Example 7: Statistical versus Practical Significance In 2003, the average age of a mother at the time of her first childbirth was To determine if the average age has increased, a random sample of 1200 mothers is taken and is found to have a sample mean age of Assuming a standard deviation of 4.8, determine whether the mean age has increased using a significance level of  =0.05.

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 1: To determine whether the mean age has increased, this is a right-tailed test with H 0 :  =25.2 versus H 1 :  >25.2. Step 2: The level of significance is  =0.05. Step 3: Recall that the sample mean is The test statistic is then

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 4: Since this is a right-tailed test, P-value = P(Z > 2.17) = The probability of obtaining a sample mean that is more than 2.17 standard deviations above the assumed mean of 25.2 is Step 5: Because the P-value is less than the level of significance, 0.05, we reject the null hypothesis.

© 2010 Pearson Prentice Hall. All rights reserved Solution Step 6: There is sufficient evidence at the 0.05 significance level to conclude that the mean age of a mother at the time of her first childbirth is greater than Although we found the difference in age to be significant, there is really no practical significance in the age difference (25.2 versus 25.5). Large sample sizes can lead to statistically significant results while the difference between the statistic and parameter is not enough to be considered practically significant.