1. Homework due Wed 2. Project Proposal due Friday Submit half-page document: Describe the question that you want to investigate. i.e population parameter of interest. Describe the data that you obtained. If found online, provide the link 3. Quiz on Friday
Last time: Hypothesis testing for mean Last time: Hypothesis testing for mean when population variance is known Today: Hypothesis testing for mean when population variance is unknown
State the null and alternative hypothesis H0: = 4, H1: ≠ 4 Review example: Medical researchers have developed a new artificial heart constructed primarily of titanium and plastic. The heart will last and operate almost indefinitely once it is implanted in the patient’s body, but the battery pack needs to be recharged about every 4 hours. We want to know if the mean battery life exceeds 4 hours. Exceeds 4 hours State the null and alternative hypothesis H0: = 4, H1: ≠ 4 H0: = 4, H1: > 4 H0: p = 4, H1: p ≠ 4 H0: p = 4, H1: p > 4
Which picture corresponds to our test? Review example: H0: = 4, H1: > 4 Which picture corresponds to our test? A) B) C)
The standard error for the sample mean can be calculated by Review example: H0: = 4, H1: > 4 A random sample of 50 battery packs is selected and subjected to a life test. The average life of these batteries is 4.05 hours. Assume that battery life is normally distributed with standard deviation = 0.2 hour. The standard error for the sample mean can be calculated by 0.2 0.2 / 𝑠𝑞𝑟𝑡 50
What is the distribution of test statistic under the null Review example: H0: = 4, H1: > 4 A random sample of 50 battery packs is selected and subjected to a life test. The average life of these batteries is 4.05 hours. Assume that battery life is normally distributed with standard deviation = 0.2 hour. What is the distribution of test statistic under the null 0.2 / 𝑠𝑞𝑟𝑡 50 From H0 From data
Compute the test statistic Review example: H0: = 4, H1: > 4 A random sample of 50 battery packs is selected and subjected to a life test. The average life of these batteries is 4.05 hours. Assume that battery life is normally distributed with standard deviation = 0.2 hour. Compute the test statistic = -1.77 = 1.77
Find p-value for the computed test statistic Review example: H0: = 4, H1: > 4 Find p-value for the computed test statistic = 1.77 0.038 0.962
p-value > Do not Reject H0 Review example: H0: = 4, H1: > 4 For p-value = 0.038. Do we reject or fail to reject the null, under significance level of = 0.05. Reject Fail to reject p-value < Reject H0 p-value > Do not Reject H0
We can construct the rejection region to reach the same conclusion Review example: Alpha = 0.05 H0: = 4, H1: > 4 We can construct the rejection region to reach the same conclusion Rejection region is (1.65, infinity ) (0.9505, infinity ) Since = 1.77 falls in the rejection region. We reject the null.
We can also construct the confidence interval. Note here is one-sided. Review example: H0: = 4, H1: > 4 We can also construct the confidence interval. Note here is one-sided. Lower bound = LB = 4.05 – 1.65* CI = (4.003, ) 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