WARM - UP The following represents snowfall amounts for 8 random winters in Detroit. 78” 130” 140” 120” 108” 120” 156” 101” 1. Fill in all four variables for the one sample 80% Confidence Interval formula: 2. What is the Margin of Error? 3. Have the Assumptions been met? SRS – Stated Approximately Normal Distribution - Stated ? - n> 30 = CLT ? - Graph ?

CHAPTER 24 - COMPARING TWO MEANS The Two Sample t-Procedures Comparing two means is the most common situation in statistical practice. The Two Sample t-Procedures The Parameters and Statistics: μ1 = the true mean response in the first group or population . μ2 = the true mean response in the second group or population. and = the mean and standard deviation from each sample. The Conditions: - The set of data must have been collected randomly. . - The Data of both groups must be Approximately Normal. - The two samples must be 100% independent of each other

Margin of Error

Critical Value t *= from the t-distribution Table The Two Sample t – Confidence Interval The Mechanics: -t* 0 t* C% The Conservative Degree of freedom: df = n(smallest) – 1 Critical Value t *= from the t-distribution Table “We can be C% confident that the true Difference in the means of the two populations is between a and b.”

Just Say NO to Pooled SRS – Both Stated Is it a good idea to listen to music when studying for a test? In a study, 49 random student were assigned to listen to Rap or Mozart while attempting to memorize many objects pictured on a page. Find the 95% Confidence Interval for the mean difference in the number of objects each group could recall. Rap Mozart Mean 10.72 11.81 S.Dev. 3.99 3.19 n 29 20 TWO Sample t – Conf. Int. SRS – Both Stated Approximately Normal Distribution – NO because both sample sizes are less than 30 for [Central Limit Theorem] = FAILS Independent We can be 95% confident that the true difference in the mean number of objects students can recall listening to Rap(μ1) – Mozart(μ2) is between -3.16 and 0.98037. Just Say NO to Pooled

NO – Zero IS in the interval = Rap(μ1) – Mozart(μ2) = 0 → μ1 = μ2 Mean 10.72 11.81 S.Dev. 3.99 3.19 n 29 20 TWO Sample t – Conf. Int. We can be 95% confident that the true difference in the mean number of objects students can recall listening to Rap(μ1) – Mozart(μ2) is between -3.16 and 0.98037. Is there evidence that Mozart listeners recalled MORE objects then Rap listeners? NO – Zero IS in the interval = Rap(μ1) – Mozart(μ2) = 0 → μ1 = μ2 IF Zero was NOT in the interval = Rap(μ1) – Mozart(μ2) ≠ 0 → μ1 ≠ μ2

HW - PAGE 566: 1, 2, 5, 6 Zero in interval = No significant difference. Zero NOT in interval significant difference.

In Repeated sampling…

t – Confidence Interval WARM - UP I want to estimate the average amount of snow fall in Buffalo, NY. Use an SRS of the following 8 Buffalo winters to set up the EQUATION used to construct a 95% Confidence Interval. Do not find the actual interval. 78” 130” 140” 120” 108” 120” 156” 101” One Sample t – Confidence Interval

Warm – Up “Freshman – 15” An average incoming freshman weighs 120lbs. Many people believe that students gain a significant amount of weight their freshman year of college. Is there enough evidence at the 5% level to support that there is an increase in weight? Use the weights of 6 randomly chosen students measured at the end of the 1st semester. μ = The true mean Weight in pounds of students at the end of 1st Semester. End 1 168 2 111 3 136 4 119 5 155 6 106 H0: μ = 120 Ha: μ > 120 ONE Sample t – Test. Since the P-Value > α = 0.05 there is NO evidence to REJECT H0 . The belief that Freshman gain weight is not supported. SRS – Stated Approximately Normal Distribution – Graph FAILED