1. Stat 301 – Day 26 Prediction Intervals Paired t-test

2. Last Time – One-Sample t Test Defining population mean  H 0 :   H a :  ≠   Test statistic: Confidence Interval for  Technical conditions  Simple random sample from population of interest  Normal population (check sample) or large sample size n Compare to T n-1

3. PP 4.4.4 (p. 364) Let  represent the mean body temperature of all 18-40 year olds adults. H 0 :  = 98.6 (the mean body temperature for adults is 98.6 o F) H a :  ≠ 98.6 (the mean body temperature is not 98.6 o F)

4. PP 4.4.4 (p. 364) The sample size is large (130 > 30) plus sample itself looks reasonably normal so quite possible population is normal  We can assume the sampling distribution of sample means is approximately normal and test statistic follows a T distribution with 29 degrees of freedom Not really a random sample  If willing to consider the sample representative of the larger population (maybe just east coasters, volunteers) then can generalize to larger population

5. PP 4.4.4 (p. 364) “Sampling Distribution” 1-Sample t With a test statistic of -5.45 and a very small p- value, we reject the null hypothesis. We have very strong evidence that the population mean body temperature differs from 98.6 o F.  Test of mu = 98.6 vs not = 98.6 Variable N Mean StDev SE Mean 95% CI T P body temp 130 98.2492 0.7332 0.0643 (98.1220, 98.3765) -5.45 0.000

6. PP 4.4.4 (p. 364) (c) A Type I Error would be concluding that the population mean body temperatures differs from 98.6 when really it does not. A Type II Error would be concluding that  is 98.6 o F when really it is different from 98.6.

7. Prediction Intervals (p. 356) NBA scoring  Saw only about 52% of games were in the confidence interval (n) What is your point estimate for the outcome of one game?  Still the sample mean (o) How far expect to be from an individual observation? How far expect to be from population mean? How far expect individual to be from  ? /n/n 

8. 90% prediction interval First get critical value for df = 24 and 90% confidence

9. 90% prediction interval 195.88 + 1.71(20.27)  (1+1/25) 195.88 + 35.35 (160.5, 231.2)  We are 90% confident that the points scored in a future game will be between 160.5 and 231.2 points.

10. Prediction Intervals Confidence interval specifies plausible values of population mean  A Prediction Interval specifies plausible values for an individual observation  Technical conditions: Simple random sample and normal population Can’t really get around it this time  Must be done “by hand”

11. Small Brainstorm Given these formulas… What are some strategies we can use to make it “easier” to find significant results/increase the precision of our estimates of the population mean?

12. Investigation 4.4.4 (p. 360) Scolari’s vs. Lucky’s  Randomly selected 29 products  Same brand, size at both stores

13. Investigation 4.4.4 (p. 360) Scolari’s vs. Lucky’s Excedrin, coffee Avg = 2.41 Avg = 2.59 Gold Medal Flour Milk 1/2G, 1G

14. Comparison Shopping ( i) H 0 :  = 0 (no tendency for one store to be more expensive) H a :  < 0 (on average, higher prices at Scolaris) (j) Test of mu = 0 vs < 0 95% Upper Variable N Mean StDev SE Mean Bound T P diffs 28 -0.118214 0.358774 0.067802 -0.002728 -1.74 0.046 We would reject the null hypothesis at the 10% level (p- value =.046 <.10). There is moderate evidence that, on average, Scolari’s has more expensive products. Variable N Mean StDev SE Mean 90% CI diffs 28 -0.118214 0.358774 0.067802 (-0.233701, -0.002728) We are 90% confident that the average price difference is between.3 cents and 23 cents (more expensive at Scolari’s).

15. The Moral By pairing items, we take out a source of random variation (e.g., different items) to help us better focus on the comparison (between stores) that we are interested in…

16. For Tuesday PP 4.4.4(d) Questions on material For Wednesday  Missing information from review handout?  Submit at least one question on material and one example exam question I could ask Thursday  If you said yes to Thursday 11-12, you are in!