1. CONFIDENCE INTERVAL OF THE MEAN, INDEPENDENT-, AND PAIRED-SAMPLES T-TESTS

2. CONFIDENCE INTERVAL OF THE MEAN We often use limited observations (samples) to talk about or estimate the population values from which they come. For example: My driver friend wants to know if he should take the F train; approximately how frequently does this train arrive? If I tell him approximately 8 minutes, how good is this estimate? How justified am I in using my SAMPLE mean here? 5 min 12 min 7 min 2 min 6 min 16 min Confidence Interval of the Mean (95% or 99%)

3. CONFIDENCE INTERVAL OF THE MEAN 5 min 12 min 7 min 2 min 6 min 16 min 95 % Confidence Interval of the Mean 8

5. CONFIDENCE INTERVAL OF THE MEAN 5 min 12 min 7 min 2 min 6 min 16 min 95 % Confidence Interval of the Mean 8+ 2.57(5.10/√6)= 13.35 8 - 2.57(5.10/√6)= 2.65 There is a 95% probability that the TRUE population mean is between 2.65 and 13.35 minutes.

6. CONFIDENCE INTERVAL OF THE MEAN 5 min 12 min 7 min 2 min 6 min 16 min 99% Confidence Interval of the Mean

8. CONFIDENCE INTERVAL OF THE MEAN 5 min 12 min 7 min 2 min 6 min 16 min 99% Confidence Interval of the Mean 8+ 4.03(5.10/√6)= 16.39 8 - 4.03(5.10/√6)= -.39 There is a 99% probability that the TRUE population mean is between -.39 and 16.39 minutes.

9. Confidence Interval of the Mean sample mean t critical value (look up in table) If 95% CI, use  =.05 If 99% CI, use  =.01 Remember, df = N-1 Standard error

10. Paired-Samples T-test Experimental design: One group, experiencing both treatments.

11. Seven people are recruited to test Proactiv acne treatment. Each person’s face is examined by a dermatologist who reports the number of pimples on each person’s face. Individuals are then instructed to use the Proactiv system of products for 3 months, after which they return to have their face pimples counted again. Test the hypothesis that Proactiv produces a difference in pimple number, using an alpha level of.05. BeforeAfter 57 66 79 55 66 79 55 H0: Proactiv does not produce a difference in pimple number. H1: Proactiv produces a difference in pimple number. Two-tailed, alpha.05, df = 6 tcrit = -2.45 and +2.45 Step 1: State the null and alternative hypotheses: Step 2: Find the critical value.

12. Seven people are recruited to test Proactiv acne treatment. Each person’s face is examined by a dermatologist who reports the number of pimples on each person’s face. Individuals are then instructed to use the Proactiv system of products for 3 months, after which they return to have their face pimples counted again. Test the hypothesis that Proactiv produces a difference in pimple number, using an alpha level of.05. BeforeAfter 57 66 79 55 66 79 55 D -2 0 0 0 0 Step 3: Calculate the obtained statistic: = -.88 ____

13. Seven people are recruited to test Proactiv acne treatment. Each person’s face is examined by a dermatologist who reports the number of pimples on each person’s face. Individuals are then instructed to use the Proactiv system of products for 3 months, after which they return to have their face pimples counted again. Test the hypothesis that Proactiv produces a difference in pimple number, using an alpha level of.05. BeforeAfter 57 66 79 55 66 79 55 D -2 0 0 0 0 Step 3: Calculate the obtained statistic: = -.88 ____.40 = - 2.15 Step 4: Make a decision. -2.45 I 2.45 I Retain the null hypothesis.

14. Independent-Samples T-test Experimental design: Two separate groups, each experiencing a different treatment.

15. A food writer would like to review the pricing of cocktails in big cities. She is looking specifically to compare the price of cocktails in Boston and New York to examine whether or not the average cocktail price is different. She goes to 7 bars in Boston and 7 bars in New York, recording the price of each bar’s Cosmopolitan. Below is the data. Test the hypothesis that Boston and New York charge significantly different prices for cocktails using an alpha level of.05. BostonNY 57 66 79 55 66 79 55 H0: Boston and NY do not charge different prices for cocktails. H1: Boston and NY do charge different prices for cocktails. For an independent-groups t-test, we use df = N-2 tcrit = -2.18 and +2.18 Alpha =.05, 2-tailed, df = 12 Step 1: State the null and alternative hypotheses: Step 2: Find the critical value.

16. A food writer would like to review the pricing of cocktails in big cities. She is looking specifically to compare the price of cocktails in Boston and New York to examine whether or not the average cocktail price is different. She goes to 7 bars in Boston and 7 bars in New York, recording the price of each bar’s Cosmopolitan. Below is the data. Test the hypothesis that Boston and New York charge significantly different prices for cocktails using an alpha level of.05. BostonNY 57 66 79 55 66 79 55 Step 3: Calculate the obtained statistic 5.86 6.71 4.86 17.43 = 5.86 – 6.71 __________

17. A food writer would like to review the pricing of cocktails in big cities. She is looking specifically to compare the price of cocktails in Boston and New York to examine whether or not the average cocktail price is different. She goes to 7 bars in Boston and 7 bars in New York, recording the price of each bar’s Cosmopolitan. Below is the data. Test the hypothesis that Boston and New York charge significantly different prices for cocktails using an alpha level of.05. BostonNY 57 66 79 55 66 79 55 Step 3: Calculate the obtained statistic Step 4: Make a decision 5.86 6.71 4.86 17.43 = 5.86 – 6.71 __________ -2.18 I 2.18 I.73 = -1.16 Retain the null hypothesis.

18. So far we have learned how to do five types of hypothesis tests: TestStatisticdfUsed when comparing:Note: Sign testOutcome (given in question) N/AComparing outcome to binomial distribution Coin flip; Number of pluses/minuses z-testzN/A to µ when σ is known Comparing sample mean to pop mean when pop sd known One sample t-test tN-1 to µ when σ is unknown Comparing sample mean to pop mean when pop sd is not known Paired- Samples t- test tN-1 to Comparing two sample means when they are from the same group of people Independent -samples t-test tN-2 to Comparing two sample means when they come from different groups of people