We’ve learned: 1) Hypothesis testing for population mean when population variance is known ( Z-test ) ( large sample size or assume population is normal ) 2) Hypothesis testing for population mean when population variance is unknown ( T-test ) ( large sample size or assume population is normal ) 3) Hypothesis testing for population proportion ( Z-test ) ( need large sample size )

What is the parameter of interest?  p Review Questions: Boys of a certain age are known to have a mean weight of 85 pounds. A complaint is made that the boys living in a municipal children's home are underfed. As one bit of evidence, n = 25 boys (of the same age) are weighed and found to have a mean weight of 80.94 pounds. It is known that the population standard deviation σ is 11.6 pounds (the unrealistic part of this example!).  Based on the available data, what should be concluded concerning the complaint? What is the parameter of interest?  p

Review Questions: What is the null and alternative hypothesis? Boys of a certain age are known to have a mean weight of 85 pounds. A complaint is made that the boys living in a municipal children's home are underfed. As one bit of evidence, n = 25 boys (of the same age) are weighed and found to have a mean weight of 80.94 pounds. It is known that the population standard deviation σ is 11.6 pounds (the unrealistic part of this example!).  Based on the available data, what should be concluded concerning the complaint? What is the null and alternative hypothesis? H0:  = 85 , H1:  ≠ 85 H0:  = 85 , H1:  < 85 H0: p = 85 , H1: p ≠ 85 H0: p = 85 , H1: p > 85

What is the test statistics under the null? a) b) c) d) Review Questions: Boys of a certain age are known to have a mean weight of 85 pounds. A complaint is made that the boys living in a municipal children's home are underfed. As one bit of evidence, n = 25 boys (of the same age) are weighed and found to have a mean weight of 80.94 pounds. It is known that the population standard deviation σ is 11.6 pounds (the unrealistic part of this example!).  Based on the available data, what should be concluded concerning the complaint? What is the test statistics under the null? a) b) c) d) 𝑍 0 = 𝑋 −  0 𝜎/ 𝑛 𝑍 0 = 𝑋 −𝜇 𝜎/ 𝑛 𝑍 0 = 𝑋 −  0 𝑠/ 𝑛 𝑇 0 = 𝑋 −  0 𝑠/ 𝑛

Which picture corresponds to our test? Review Questions: H0:  = 85 , H1:  < 85 Which picture corresponds to our test? A) B) C)

What is the value of the test statistic? a) −1.75 b) −1.65 Review Questions: Boys of a certain age are known to have a mean weight of 85 pounds. A complaint is made that the boys living in a municipal children's home are underfed. As one bit of evidence, n = 25 boys (of the same age) are weighed and found to have a mean weight of 80.94 pounds. It is known that the population standard deviation σ is 11.6 pounds (the unrealistic part of this example!).  Based on the available data, what should be concluded concerning the complaint? 𝑍 0 = 𝑋 −  0 𝜎/ 𝑛 What is  the value of the test statistic? a) −1.75 b) −1.65

p-value >   Do not Reject H0 Review Questions : H0:  = 85 , H1:  < 85 Find p-value for the computed test statistic = −1.75 0.04 0.96 p-value <   Reject H0 p-value >   Do not Reject H0

The p-value is P(Z < −1.75) = 0.04, Under significance level 0.05 Review example: H0:  = 85 , H1:  < 85 The p-value is P(Z < −1.75) = 0.04, Under significance level 0.05 Reject Fail to reject

We can construct the rejection region to reach the same conclusion Review Questions : 𝛼 = 0.05 H0:  = 85 , H1:  < 85 We can construct the rejection region to reach the same conclusion Rejection region is (1.65, ∞) (- ∞ , -1.65) − 𝑍 𝛼 Since = -1.75 falls in the rejection region. We reject the null. Reach the same conclusion

We can also construct the confidence interval. Note here is one-sided. Review Questions : H0:  = 85 , H1:  < 85 We can also construct the confidence interval. Note here is one-sided. Upper bound = CI = ( − ∞ , 84.76 ) CI = ( 84.76 , ∞ ) 𝑋 + 𝑍 0.05 𝜎 𝑛 With 95% confidence level, the population mean lies in the confident interval Null value is not in the 95% confidence interval. We can reject the null hypothesis

Review Questions: a) b) c) d) Boys of a certain age are known to have a mean weight of 85 pounds. A complaint is made that the boys living in a municipal children's home are underfed. As one bit of evidence, n = 25 boys (of the same age) are weighed and found to have a mean weight of 80.94 pounds. It is known that the population standard deviation σ is 11.6 pounds (the unrealistic part of this example!).  Based on the available data, what should be concluded concerning the complaint? If we assume σ is unknown. What is the test statistics under the null? a) b) c) d) 𝑍 0 = 𝑋 −  0 𝜎/ 𝑛 𝑍 0 = 𝑋 −𝜇 𝜎/ 𝑛 𝑍 0 = 𝑋 −  0 𝑠/ 𝑛 𝑇 0 = 𝑋 −  0 𝑠/ 𝑛

Review Questions: Boys of a certain age are known to have a mean weight of 85 pounds. A complaint is made that the boys living in a municipal children's home are underfed. As one bit of evidence, n = 25 boys (of the same age) are weighed and found to have a mean weight of 80.94 pounds. It is known that the population standard deviation σ is 11.6 pounds (the unrealistic part of this example!).  Based on the available data, what should be concluded concerning the complaint? If we assume σ is unknown. What is the test statistics under the null? Z distribution T distribution with df = 25 T distribution with df = 24 𝑇 0 = 𝑋 −  0 𝑠/ 𝑛

Review Questions: Writing Hypotheses Are more than 80% of American’s right handed? H0: p = 0.8 , H1: p > 0.8 H0: p = 0.8, H1: p >= 0.8 H0: p = 0.8, H1: p <= 0.8 H0: p = 0.8, H1: p < 0.8

Review Questions: Writing Hypotheses Is the proportion of babies born male different from .50? H0: p = 0.5, H1: p ≠ 0.5 H0: p = 0.5, H1: p > 0.5 H0: p = 0.5, H1: p < 0.5

Review Questions: Writing Hypotheses Is the percentage of Creamery customers who prefer chocolate ice cream over vanilla less than 80%? H0: p = 0.8 , H1: p > 0.8 H0: p = 0.8, H1: p >= 0.8 H0: p = 0.8, H1: p <= 0.8 H0: p = 0.8, H1: p < 0.8

Hypothesis Testing on a Binomial Proportion

Are you ready for something different? What if we are more interested in testing the variation (spread) in our population distribution. This is especially in quality control. For example, we want a coffee dispenser to fill up a cup of coffee with a standard deviation less than 0.02 oz How do we test if the coffee dispenser satisfy our expectation We want to test the population variance 𝜎 2