Student Learning Outcomes The student will explore hypothesis testing with single mean and known population standard deviation data.

Given Suppose that a recent article stated that the average time spent in jail by a first–time convicted burglar is 2.5 years. A study was then done to see if the average time has increased in the new century. A random sample of 26 first–time convicted burglars in a recent year was picked. The average length of time in jail from the survey was 3 years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the average length of jail time has increased.

Hypothesis Testing: Single Average Is this a test of averages or proportions? Averages State the null and alternative hypotheses. aHo size 12{H rSub { size 8{O} } } {}: b Ha size 12{H rSub { size 8{a} } } {}: a Ho:μ=2. 5 size 12{H rSub { size 8{O} } :μ=2 "." 5} {} (or, Ho:μ≤2.5 size 12{H rSub { size 8{O} } :μ <= 2 "." 5} {}) b H a : μ > 2 . 5 size 12{H rSub { size 8{a} } :μ>2 "." 5} {} Is this a right-tailed, left-tailed, or two-tailed test? How do you know? right-tailed What symbol represents the Random Variable for this test? X ¯ size 12{ {overline {X}} } {} In words, define the Random Variable for this test. The average time spent in jail for 26 first time convicted burglars Is the population standard deviation known and, if so, what is it? Yes, 1.5 Calculate the following: a x¯= size 12{ {overline {x}} ={}} {} b σ= size 12{σ={}} {} c sx= size 12{s rSub { size 8{x} } ={}} {} d n= size 12{n={}} {} a3 b1.5 c1.8 d26 Since both σ size 12{σ} {} and sx size 12{s rSub { size 8{x} } } {} are given, which should be used? In 1 -2 complete sentences, explain why. σ size 12{σ} {} State the distribution to use for the hypothesis test. X ¯ ~ N ( 2 . 5 , 1.5 26 ) size 12{ {overline {X}} "~" N \( 2 "." 5,1 "." 5 sqrt {"26"} \) } {} Sketch a graph of the situation. Label the horizontal axis. Mark the hypothesized mean and the sample mean x¯ size 12{ {overline {x}} } {}. Shade the area corresponding to the p-value. Find the p-value. 0.0446 At a pre-conceived α=0.05 size 12{α=0 "." "05"} {}, what is your: a Decision: b Reason for the decision: c Conclusion (write out in a complete sentence): aReject the null hypothesis

Discussion Questions Does it appear that the average jail time spent for first time convicted burglars has increased? Why or why not?