bowerman_9e_chap_102.pptx

Chapter 10
Hypothesis Testing

Copyright ©2018 McGraw-Hill Education. All rights reserved.

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Chapter Outline
10.1 The Null and Alternative Hypotheses and Errors in Hypothesis Testing
10.2 Tests about a Population Mean: Known
10.3 Tests about a Population Mean: Unknown
10.4 Tests about a Population Proportion
10.5 Type II Error Probabilities and Sample Size Determination (Optional)
10.6 The Chi-Square Distribution
10.7 Statistical Inference for a Population Variance (Optional)
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10.1 Null and Alternative Hypotheses and Errors in Hypothesis Testing
One-Sided, “Greater Than” Alternative
H0: μ  μ0 vs. Ha: μ > μ0
One-Sided, “Less Than” Alternative
H0 : μ  μ0 vs. Ha : μ < μ0 Two-Sided, “Not Equal To” Alternative H0 : μ = μ0 vs. Ha : μ  μ0 where μ0 is a given constant value (with the appropriate units) that is a comparative value LO10-1: Set up appropriate null and alternative hypotheses. 10-3 3 The Idea of a Test Statistic LO10-1 10-4 4 Type I and Type II Errors Table 10.1 LO10-2: Describe Type I and Type II errors and their probabilities. 10-5 5 Typical Values Low alpha gives small chance of rejecting a true H0 Typically,  = 0.05 Strong evidence is required to reject H0 Usually choose  between 0.01 and 0.05  = 0.01 requires very strong evidence is to reject H0 Tradeoff between  and β For fixed n, the lower , the higher β LO10-2 10-6 6 10.2 Tests about a Population Mean: Known Test hypotheses about a population mean using the normal distribution Called tests Require that the true value of the population standard deviation is known In most real-world situations, is not known When is unknown, test hypotheses about a population mean using the distribution Here, assume that we know LO10-3: Use critical values and p-values to perform a test about a population mean when is known. 10-7 7 Steps in Testing a “Greater Than” Alternative State the null and alternative hypotheses Specify the significance level alpha () Select the test statistic Determine the critical value rule for rejecting 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 critical value rule Interpret the statistical results in managerial terms and assess their practical importance LO10-3 10-8 8 Steps in Testing a “Greater Than” Alternative in Trash Bag Case #1 State the null and alternative hypotheses H0:   50 Ha:  > 50
Specify the significance level  = 0.05
Select the test statistic
LO10-3
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Steps in Testing a “Greater Than”
Alternative in Trash Bag Case #2
Determine the critical value rule for deciding whether or not to reject H0
Reject H0 in favor of Ha if the test statistic is greater than the rejection point 
This is the critical value rule
In the trash bag case, the critical value rule is to reject H0 if the calculated test statistic is > 1.645
LO10-3
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LO10-3
Steps in Testing a “Greater Than”
Alternative in Trash Bag Case #3
Figures 10.1 (partial) and 10.2 (partial)
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Steps in Testing a “Greater Than”
Alternative in Trash Bag Case #4
Decide whether to reject H0 by using the test statistic and the rejection rule
Compare the value of the test statistic to the critical value according to the critical value rule
In the trash bag case, = 2.20 is greater than 0.05 = 1.645
Therefore reject H0: μ ≤ 50 in favor of
Ha: μ > 50 at the 0.05 significance level
Interpret the statistical results in managerial terms and assess their practical importance
LO10-3
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Effect of 
At  = 0.01, the rejection point is 0.01 = 2.33
In the trash example, the test statistic
= 2.20 is < 0.01 = 2.33 Therefore, cannot reject H0 in favor of Ha at the  = 0.01 significance level This is the opposite conclusion reached with  = 0.05 So, the smaller we set , the larger is the rejection point, and the stronger is the statistical evidence that is required to reject the null hypothesis H0 LO10-3 10-13 13 The p-Value The p-value is the probability of the obtaining the sample results if the null hypothesis H0 is true Sample results that are not likely if H0 is true have a low p-value and are evidence that H0 is not true The p-value is the smallest value of  for which we can reject H0 The p-value is an alternative to testing with a test statistic LO10-3 10-14 14 Steps Using a p-value to Test a “Greater Than” Alternative LO10-3 10-15 15 Steps in Testing a “Less Than” Alternative in Payment Time Case #1 State the null and alternative hypotheses H0:  ≥ 19.5 vs. Ha:  < 19.5 Specify the significance level  = 0.01 Select the test statistic LO10-3 10-16 16 Steps in Testing a “Less Than” Alternative in Payment Time Case #2 Determine the rejection rule for deciding whether or not to reject H0 The rejection rule is to reject H0 if the calculated test statistic – is less than –2.33 Collect the sample data and calculate the value of the test statistic LO10-3 10-17 17 Steps in Testing a “Less Than” Alternative in Payment Time Case #3 Decide whether to reject H0 by using the test statistic and the rejection rule In the payment time case, = –2.67 is less than 0.01 = –2.33 Therefore reject H0: μ ≥ 19.5 in favor of Ha: μ < 19.5 at the 0.01 significance level Interpret the statistical results in managerial terms and assess their practical importance LO10-3 10-18 18 Steps Using a p-value to Test a “Less Than” Alternative Collect the sample data, compute the value of the test statistic, and calculate the p‑value by corresponding to the test statistic value Reject H0 if the p-value is less than  LO10-3 10-19 19 Steps in Testing a “Not Equal To” Alternative in Valentine Day Case #1 State null and alternative hypotheses H0:  = 330 vs. Ha:  ≠ 330 Specify the significance level  = 0.05 Select the test statistic LO10-3 10-20 20 Steps in Testing a “Not Equal To” Alternative in Valentine Day Case #2 Determine the rejection rule for deciding whether or not to reject H0 Rejection points are  =1.96, – = – 1.96 Reject H0 in favor of Ha if the test statistic satisfies either: greater than the rejection point  /2, or – less than the rejection point – /2 LO10-3 10-21 21 Steps in Testing a “Not Equal To” Alternative in Valentine Day Case #3 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 LO10-3 10-22 22 Steps Using a p-value to Test a “Not Equal To” Alternative Collect the sample data and compute the value of the test statistic Calculate the p-value by corresponding to the test statistic value The p-value is 0.1587 · 2 = 0.3174 Reject H0 if the p-value is less than  LO10-3 10-23 23 Interpreting the Weight of Evidence Against the Null Hypothesis If p < 0.10, there is some evidence to reject H0 If p < 0.05, there is strong evidence to reject H0 If p < 0.01, there is very strong evidence to reject H0 If p < 0.001, there is extremely strong evidence to reject H0 LO10-3 10-24 24 10.3 Tests about a Population Mean: Unknown Suppose the population being sampled is normally distributed The population standard deviation is unknown, as is the usual situation If the population standard deviation is unknown, then it will have to estimated from a sample standard deviation Under these two conditions, the distribution must be used to test hypotheses LO10-4: Use critical values and p-values to perform a test about a population mean when is unknown. 10-25 25 Defining the Random Variable: Unknown Define a new random variable The sampling distribution of this random variable is a distribution with n – 1 degrees of freedom LO10-4 10-26 26 Defining the Statistic: Unknown Let 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 If the population being sampled is normal, and is used to estimate , then … The sampling distribution of the statistic is a distribution with n – 1 degrees of freedom LO10-4 10-27 27 Tests about a Population Mean: Unknown LO10-4 10-28 28 10.4 Tests about a Population Proportion LO 5: Use critical values and p-values to perform a large sample test about a population proportion. 10-29 29 Example 10.6 The Cheese Spread Case: Improving Profitability LO10-5 10-30 30 10.5 Type II Error Probabilities and Sample Size Determination (Optional) Compute the probability β of not rejecting a false null hypothesis That is, compute the probability β of committing a Type II error 1 - β is called the power of the test LO10-6: Calculate Type II error probabilities and the power of a test, and determine sample size (Optional). 10-31 31 Calculating β Assume that the sampled population is normally distributed, or that a large sample is taken Test… H0: µ = µ0 vs Ha: µ < µ0 or Ha: µ > µ0 or Ha: µ ≠ µ0
Set the probability of a Type I error equal to  and randomly select a sample of size n
LO10-6
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Calculating β Continued
Probability β of a Type II error corresponding to the alternative value µa for µ is the area under the standard normal curve to the left of
Here * equals  if the alternative hypothesis is one-sided (µ < µ0 or µ > µ0)
Also * ≠ /2 if the alternative hypothesis is two-sided (µ ≠ µ0)
LO10-6
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Sample Size
Assume the sampled population is normally distributed, or that a large sample is taken
Test 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  and the probability of a Type II error corresponding to the alternative value μ for μ equal to β
Sample size is
LO10-6
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10.6 The Chi-Square Distribution (Optional)
Figures 10.9 and 10.10
LO10-7: Describe the properties of the chi-square distribution and use a chi-square table (Optional).
The chi-square ² distribution depends on the number of degrees of freedom
A chi-square point ² is the point under a chi-square distribution that gives right-hand tail area 

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10.7 Statistical Inference for a Population Variance (Optional)
If 2 is the variance of a random sample of n measurements from a normal population with variance 2
The sampling distribution of the statistic
is a chi-square distribution with (n – 1) degrees of freedom
Can calculate confidence interval and perform hypothesis testing
LO10-8: Use the chi-square distribution to make statistical inferences about a population variance (Optional).
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Confidence Interval for Population Variance

LO10-8
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Statistical Inference for a Population Variance

LO10-8
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Selecting an Appropriate Test Statistic to Test a Hypothesis about a Population Mean
Figure 10.13

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