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3. Estimating the Population Mean with Confidence One-sample t Confidence Interval for  where t* is from a t(n – 1) distribution. Interval requires have a r.s. from normal popul. If sample size large (n > 30), assumption of normality not so crucial and result is approx.

4. CI for Mean Maximum Distance: SPSS output

5. 13.1 Intro to Testing for Means pg 123 Page 551 three notes of caution: 1.Inference only valid if sample is representative of population for question of interest. 2.Hypotheses/conclusions apply to larger population(s) represented by the sample(s). 3.If distribution of quant. variable is highly skewed, consider analyzing median rather than the mean -- nonparametric methods.

6. Chpt 13 Testing about Means Basic Steps: 1:Determine null and alternative hypotheses. 2:Verify necessary data conditions, if met, summarize data into appropriate test statistic. For proportions, the standardized statistic is called a _______________ and the _________________ used to find p-value. For means, the standardized statistic is called a _______________ and the _____________ used to find p-value.

7. Chpt 13 Testing about Means Basic Steps continued: 3:Assuming null hypothesis is true, find the p-value. 4:Decide if result is statistically significant based on p-value. 5:Report conclusion in context of the situation.

8. The Beauty of p-values Suppose the significance level  is set at 5% for testing H 0 : status quo versus H a : the “new theory”. If p-value is… Statistical Decision Feasible Conclusion about the “New Theory” 0.462 Fail to Reject H 0 0.063 Fail to Reject H 0 0.041Reject H 0 0.003Reject H 0 Not even close to significance – maybe throw out that “new theory” Just missed being significant at 5% -- Practically important? New study? Larger n? Significant at the 5% level -- sufficient support for the “new theory” Highly significant -- strong support for the “new theory”

9. 13.3 Testing about a Population Mean  Situation: one population and response is quantitative test about value of mean response for population  Data = random sample Response is normally distributed for population (but if n is large, this condition is less crucial). 1.H 0 :H a : 2.H 0 : H a : 3.H 0 : H a :

10. Step 2: Conditions and Test Statistic How would you check conditions? Test statistic = Sample statistic – Null value Standard error If H 0 true, this test statistic has a _______________________ distribution.

11. Step 3: Find the p-value Steps … Draw distribution for test statistic under H 0. For t tests = t-distribution with certain df Locate observed test statistic value on axis. Shade area that corresponds to p-value. Look at H a for direction of extreme. Use appropriate table find (bounds for) p-value. For t tests we will use Table A.3. From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with permission.

12. In Step 4: we need to decide if the result is Statistically Significant. i.e. see if the p-value is ≤  Step 5: Report conclusion in context of situation.

13. Page 127 From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with permission.

14. Try It! Using Table A.3 page 128 H 0 :  = 0 vs H a :  > 0 with n=15 obs and t=1.97 Draw distribution for test statistic under H 0. Locate observed test statistic value on axis. Shade area that corresponds to p-value. Use table find (bounds for) p-value. Get ready to click in your answer on next slide …

15. A = Yes or B = No click in your answer … Is the value of t = 1.97 significant at the 5% level? Is it significant at the 1% level?

16. Try It! Using Table A.3 H 0 :  = 64 vs H a :  ≠ 64 with n=30 obs and t=1.12 How would you report the p-value?

17. Try It! Classical Music page 129 Test if HS students complete a maze more quickly while listening to classical music. For general HS population, time to complete maze follows a normal distribution with a mean of 40 seconds. Use a 5% significance level. Define the parameter: Let  represent State the hypotheses: H 0 :H a :

18. Try It! Classical Music Random Sample of 100 HS students timed while listening to classical music.  mean time = 39.1 seconds  std dev = 4 seconds. Conduct test:

19. Try It! Classical Music Are results statistically significant at the 5% level? State conclusion at 5% level in terms of the problem. Comment about assumptions required for test