1. WARM – UP 1. What is the Critical Value t* for a distribution of 26 observation with probability 0.10 to the Right? 2. What is the Critical Value t* for a 98%Confidence Interval of 4 observations? 3. What is the p-value for a one-sample t-Test with H0: μ = 3 vs. Ha: μ > 3, n = 27 and t = 2.98 4. What is the p-value for a one-sample t-Test with H0: μ = 4 vs. Ha: μ ≠ 4, n = 7 and t = -1.75 1.316 4.541 0.0031 0.1307

2. WARM – UP In a study of heart surgery, one issue was the effects of drugs called beta blockers on the pulse rate of patients during surgery. Subjects were randomly divided into two groups of 30 patients each. One group received a beta blocker, the other a placebo. Pulse rates were recorded during the operations. The treatment group had a mean of 65.2 beats/min. (s = 7.8), while the control group was 70.3 beats/min. (s = 8.3). Give a 95% confidence Interval for the mean difference. Treatment:65.2 7.8 Control:70.3 8.3

3. μ 1 = The true mean number of beats/min for the treatment group. μ 2 = The true mean number of beats/min for the control group. We can be 95% confident that the true mean difference in the number of beats/min is between -9.263 and -0.937 (β Blockers – Placebo). 1.SRS – Stated 2. Independent 3. Approximately Normal Distribution – YES because both sample sizes are ≥ 30 [Central Limit Theorem] TWO Sample t – Conf. Int. Treatment:65.2 7.8 Control:70.3 8.3

4. H 0 : μ 1 = μ 2 H a : μ 1, or ≠ μ 2 The Hypothesis: The Mechanics: The t Test Statistic = Ha: μ 1 < μ 2 tcdf(-E99, t, df) Ha: μ 1 > μ 2 tcdf(t, E99, df) Ha: μ 1 ≠ μ 2 2·tcdf(|t|, E99, df) CH. 24 - The Two Sample t – Test The Conservative Degree of freedom: df = n (smallest) 1 df = n (smallest) – 1 The P-Value =

5. In a study of heart surgery, one issue was the effects of drugs called beta blockers on the pulse rate of patients during surgery. Subjects were randomly divided into two groups of 30 patients each. One group received a beta blocker the other a placebo. Pulse rates were recorded during the operations. The treatment group had a mean of 65.2 beats/min. (s = 7.8), while the control group was 70.3 beats/min. (s = 8.3). Is there evidence that beta blockers reduce pulse rate? H 0 : μ 1 = μ 2 H a : μ 1 < μ 2 Since the P-Value is less than α = 0.01 the data IS significant. There is strong evidence to REJECT H 0. Beta blockers do reduce pulse. TWO Sample t – Test μ 1 = The true mean number of beats/min for the treatment group. μ 2 = The true mean number of beats/min for the control group. 1.SRS – Stated 2. Independent 3. Approximately Normal Distribution – YES because both sample sizes are above 30 [Central Limit Theorem]

6. Does Buffalo, New York, receive significantly more snow than Chicago, Illinois? To test this question, researchers randomly collected data of yearly snowfall amounts in each city. What conclusions can be made from this data? Buffalo: 78” 130” 140” 120” 108” 120” 156” 101” Chicago: 84” 96” 114” 81” 98” 103” μ 1 = The true mean amount of annual Snow fall in Buffalo, NY. μ 2 = The true mean amount of annual Snow fall in Chicago, IL. H 0 : μ 1 = μ 2 H a : μ 1 > μ 2 Since the P-Value is less than α = 0.05 the data IS significant. There is strong evidence to REJECT H 0. Buffalo’s yearly snowfall average is greater than that of Chicago, IL 1.SRS – Both Stated 2. Independent 3. Approximately Normal Distribution – Graph BOTH! TWO Sample t – Test

7. Is there evidence to suggest that Mr. Belin’s Period 3 and Period 4 AP Statistics Classes performed differently on Quiz #1? To test this an SRS of students were selected from each class. Period 3: 86 100 98 72 100 86 98 100 Period 4: 68 94 98 82 80 83 88 100 76 92 μ i = The true mean Quiz score of Mr. Belin AP Statistic class (Period 3 and Period 4). H 0 : μ 3 = μ 4 H a : μ 3 ≠ μ 4 Since the P-Value is NOT less than α = 0.05 the data IS Not significant. There is NO evidence to REJECT H 0. Period 3 and Period 4 perform similarly on quiz #1. 1.SRS – Stated 2. Independent 3. Approximately Normal Distribution – Graph BOTH! TWO Sample t – Test

9. Page 566: #1 The Core Plus Mathematics Project (CPMP) in an innovative approach to teaching Math to kids by engaging them in group investigation and math modeling. 320 Students from the CPMP program and 273 from traditional curriculum were given an algebra test, the scores were recorded, and a 95% confidence interval was computed for the mean difference between the results: [ 5.573, 11.427 ] a.) Margin of Error? b.) Would a 98% be larger or smaller? c.) Interpret d.) Is the CPMP program better?

10. HOMEWORK: Page 566: 3, 4, 9, 10, 12, 16 #3 The Core Plus Mathematics Project (CPMP) also gave students tests and did NOT allow calculators. The results are indicated below. Are the mean scores of the 2 groups significantly different? a.) Hypothesis b.) Assumptions SRS, Appr. Norm. via the CLT (large n) c.&d.) Conclusion. CPMP:29.0 18.8 312 Traditional:38.4 16.2 265 μ 1 = The true mean score for 1=CPMP, 2=Trad. H 0 : μ 1 = μ 2 H a : μ 1 ≠ μ 2