1. Confidence Intervals and Hypothesis Tests Week 5

2. Confidence Intervals Have: A percent of confidence: how sure are we that the true proportion is in the interval. The interval: the low and high numbers that we are x% sure contain the true proportion.

3. Confidence Intervals All Confidence Intervals are an estimate plus or minus a margin of error. Estimate ± Margin of Error Margin of error is the specific number of standard errors (Std deviations) needed to surround the percent of confidence

4. How to Make a Confidence Interval for a Proportion Find, the sample proportion from the data Find SE( )= Find z* for the level of confidence:  90%  1.645, 95%  1.96, 99%  2.576 Find margin of error: ME = z*SE( ) Find interval = ± ME

5. Confidence Intervals on the Calculator STATS  TESTS  A:1-PropZInt Fill in the numbers  X: number of successes in the sample  N: sample side  C-Level: percent of confidence Calculate Read off the interval

6. What can we say? We are 95% sure that the interval between 84% and 92% captures the true proportion of all orders delivered on time. We are 90% sure that the interval from 78% to 84% contains the true proportion of people who think original Frumpies are better than new Frumpies.

7. What do we think? The interval is the subject not the proportion.  We do not know anything about the proportion. The interval is one person’s calculated guess.  Each sample will produce a different interval. The interval is not guaranteed to have captured the proportion.  We only have 90%, 95%, 99% certainty that the interval captured the proportion

8. What we cannot imply. The parameter does not move around.  95% of the time the parameter will be in the interval Other people’s intervals will be like yours  95% of all samples will have proportions that will be in the interval. The parameter is sure to be in the interval  The true proportion will be within the interval

9. Tests of Significance

10. Think Hypothesis test starts with a careful statement of the alternatives. Form a null hypothesis H 0 about a parameter, usually the mean, µ or p  H 0 must be about a parameter  Usually about what we do not want to show  Usually states that the treatment had no effect

11. Think Form an alternative hypothesis, H a  H a is what we “really want” to show  Usually that the treatment had and effect Thought pattern  Assume H 0 is true  Is the sample outcome surprisingly larger or smaller than anticipated?  If it is, then there is evidence that H 0 is false and that H a is true  Ask the question: How unlikely is this sample result if the null hypothesis is true?  P-value is a measurement of the evidence against H o. If p-value is small (less than.05) we reject H o If p-value is large (more than.05) we do not reject H o

12. Think The statement being tested is H o. The test is designed so to assess the strength of the evidence against H o. Usually the null hypothesis is a statement of no effect or no difference. H 0 : p = some value is a common example

13. Think The alternative hypothesis, H a states what we hope the evidence will show. Two sided alternative: H a : p ≠ same value  We want to show that the true mean is not the same as it was after the treatment One sided alternative: H a : p > same value or H a : p < same value  We want to show that the true mean has been increased or decreased after the treatment

14. Thought Process We assume the claim is true. We take a sample and want to see if the sample “supports” or “goes against” the claim. We do not expect the sample to exactly mirror the claim. The real world does not come out even, but how far off is two far? We will use the number of standard errors away from the claim as a measurement

15. Mechanics Find the standard deviation of p Find how many standard deviations off the given proportion is Use the Standard normal table to find the probability of a value this extreme = p value The p value is the probability that we would get a sample this far off if H o is true

16. Mechanics with the Calculator STATS  TESTS  5:1-PropZTest Fill in the numbers  Po: the claimed proportion  X: number of successes in the sample  N: sample size  Choose sample proportion ≠Po, Po  Calculate

17. Calculator (page 2) Read results:  Z = number of standard errors away from the mean  P= p-value, the measurement of how rare the sample is if the claim is true.  = the sample proportion

18. Show Sometimes before the calculations are done a set cut-off point will be established such that if the p-value is smaller than the cut-off point then H 0 will be rejected. Common levels are.05 and.01 This cut-off point is call a significance level and is denoted by the Greek letter alpha, If the p-value is as small or smaller than, we say that it is statistically significant at that level.

19. Tell After finding the p-value comment on the likelihood of getting a value as extreme as the test statistic given that H 0 is true.

20. Tell What can we say if the p-value is small  “The is strong evidence to reject H 0 that... and to accept H a that.... “ Use context  “The small likelihood of getting such a small p- value given H 0 is true leads us to reject H 0 that … and to accept H a that....

21. Tell What we can say if the p-value is large  “Based on the p-value we can not reject H 0 that...”  “There is not enough evidence for us to reject H 0 that....”  Answer in the context of the problem.

22. Confidence Interval for Means

23. How to make a C.I. for means Find SE(sample mean)  s=standard deviation of the data  n=sample size Find degrees of freedom dof = n-1 Use the T chart to find the magic number t* Find

24. Confidence Intervals on Sample means on the TI STATS  TESTS  Tinterval Choose Stats Fill in the numbers Sample mean Sx:Sample standard deviation N: Sample size C-Level: Percentage of confidence desired Calculate

25. Calculator page 2 Results: The interval  (Low number, High number) for the mean We are C-Level% sure the true mean of the population is between Low number, High number

26. Hypothesis Tests for Means

27. How do you do it Find SE(sample mean)  s=standard deviation of the data  n=sample size Find degrees of freedom dof = n-1 Find how many SEs off the µ the sample mean is:

28. How do you do it part 2 Look up the p-value in the T chart  Find the degrees of freedom row  Look up the desired cut-off points Compare the t value with the cut-off points to estimate the p-value

29. Confidence Intervals for the mean by Calculator STATS  TESTS  2: T-Test Choose STATS Enter the numbers  : Claimed value of the mean  : Sample mean  Sx: Sample standard deviation  N : Sample size  Choose ≠, Calculate

30. Calculator page 2 Results:  T: number of Standard errors off the claim  P: p-value. Proportion of the time a sample this unusual will occur if the claim is true