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 )

We’ve learned:

Review Questions:

What is the parameter of interest? )a

b) p

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?

Review Questions:

What is the null and alternative hypothesis? a) H0: = 85 , H1: ≠ 85 b) H0: = 85 , H1: < 85 c) H0: p = 85 , H1: p ≠ 85 d) H0: p = 85 , H1: p > 85

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?

Review Questions:

What is the test statistics under the null?

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?

𝑍 0=𝑋−0

𝜎 /√𝑛𝑍 0=

𝑋−𝜇𝜎 /√𝑛 𝑍 0=

𝑋−0

𝑠 /√𝑛𝑇 0=

𝑋−0

𝑠 /√𝑛

Review Questions:

Which picture corresponds to our test?

H0: = 85 , H1: < 85

A) B) C)

Review Questions:

What is the value of the test statistic?

a) −1.75 b) −1.65

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

Review Questions :

Find p-value for the computed test statistic = −1.75

a) 0.04b) 0.96

H0: = 85 , H1: < 85

p-value < Reject H0

p-value > Do not Reject H0

Review example:

The p-value is P(Z < −1.75) = 0.04, Under significance level 0.05

a) Rejectb) Fail to reject

H0: = 85 , H1: < 85

Review Questions :

We can construct the rejection region to reach the same conclusion Rejection region is a) (1.65, )b) (--1.65)

H0: = 85 , H1: < 85

Since = -1.75 falls in the rejection region. We reject the null. Reach the same conclusion

= 0.05

−𝑍𝛼

Review Questions :

We can also construct the confidence interval. Note here is one-sided.Upper bound =

a) CI = ( , 84.76 )b) CI = ( 84.76 , )

H0: = 85 , H1: < 85

• 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

𝑋+𝑍 0.05𝜎√𝑛

Review Questions:

If we assume σ is unknown. What is the test statistics under the null?

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?

𝑍 0=𝑋−0

𝜎 /√𝑛𝑍 0=

𝑋−𝜇𝜎 /√𝑛 𝑍 0=

𝑋−0

𝑠 /√𝑛𝑇 0=

𝑋−0

𝑠 /√𝑛

Review Questions:

If we assume σ is unknown. What is the test statistics under the null?

a) Z distribution b) T distribution with df = 25c) T distribution with df = 24

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?

𝑇 0=𝑋−0

𝑠 /√𝑛

Review Questions: Writing Hypotheses

a) H0: p = 0.8 , H1: p > 0.8b) H0: p = 0.8, H1: p >= 0.8c) H0: p = 0.8, H1: p <= 0.8d) H0: p = 0.8, H1: p < 0.8

Are more than 80% of American’s right handed?

Review Questions: Writing Hypotheses

a) H0: p = 0.5, H1: p ≠ 0.5b) H0: p = 0.5, H1: p > 0.5c) H0: p = 0.5, H1: p < 0.5

Is the proportion of babies born male different from .50?

Review Questions: Writing Hypotheses

Is the percentage of Creamery customers who prefer chocolate ice cream over vanilla less than 80%?

a) H0: p = 0.8 , H1: p > 0.8b) H0: p = 0.8, H1: p >= 0.8c) H0: p = 0.8, H1: p <= 0.8d) 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