1. Significance Tests About
Section 8.2
Significance Tests About
Proportions

2. Example: Are Astrologers’ Predictions Better Than Guessing?
Scientific “test of astrology” experiment:
For each of 116 adult volunteers, an astrologer prepared a horoscope based on the positions of the planets and the moon at the moment of the person’s birth
Each adult subject also filled out a California Personality Index Survey

3. Example: Are Astrologers’ Predictions Better Than Guessing?
For a given adult, his or her birth data and horoscope were shown to an astrologer together with the results of the personality survey for that adult and for two other adults randomly selected from the group
The astrologer was asked which personality chart of the 3 subjects was the correct one for that adult, based on his or her horoscope

4. Example: Are Astrologers’ Predictions Better Than Guessing?
28 astrologers were randomly chosen to take part in the experiment
The National Council for Geocosmic Research claimed that the probability of a correct guess on any given trial in the experiment was larger than 1/3, the value for random guessing

5. Example: Are Astrologers’ Predictions Better Than Guessing?
Put this investigation in the context of a significance test by stating null and alternative hypotheses

6. Example: Are Astrologers’ Predictions Better Than Guessing?
With random guessing, p = 1/3
The astrologers’ claim: p > 1/3
The hypotheses for this test:
Ho: p = 1/3
Ha: p > 1/3

7. What Are the Steps of a Significance Test about a Population Proportion?
Step 1: Assumptions
The variable is categorical
The data are obtained using randomization
The sample size is sufficiently large that the sampling distribution of the sample proportion is approximately normal:
np ≥ 15 and n(1-p) ≥ 15

8. What Are the Steps of a Significance Test about a Population Proportion?
Step 2: Hypotheses
The null hypothesis has the form:
Ho: p = po
The alternative hypothesis has the form:
Ha: p > po (one-sided test) or
Ha: p < po (one-sided test) or
Ha: p ≠ po (two-sided test)

9. What Are the Steps of a Significance Test about a Population Proportion?
Step 3: Test Statistic
The test statistic measures how far the sample proportion falls from the null hypothesis value, po, relative to what we’d expect if Ho were true
The test statistic is:

10. The P-value summarizes the evidence
What Are the Steps of a Significance Test about a Population Proportion?
Step 4: P-value
The P-value summarizes the evidence
It describes how unusual the data would be if H0 were true

11. We summarize the test by reporting and interpreting the P-value
What Are the Steps of a Significance Test about a Population Proportion?
Step 5: Conclusion
We summarize the test by reporting and interpreting the P-value

12. Example: Are Astrologers’ Predictions Better Than Guessing?
Step 1: Assumptions
The data is categorical – each prediction falls in the category “correct” or “incorrect” prediction
Each subject was identified by a random number. Subjects were randomly selected for each experiment.
np=116(1/3) > 15
n(1-p) = 116(2/3) > 15

13. Example: Are Astrologers’ Predictions Better Than Guessing?
Step 2: Hypotheses
H0: p = 1/3
Ha: p > 1/3

14. Example: Are Astrologers’ Predictions Better Than Guessing?
Step 3: Test Statistic:
In the actual experiment, the astrologers were correct with 40 of their 116 predictions (a success rate of 0.345)

15. Example: Are Astrologers’ Predictions Better Than Guessing?
Step 4: P-value
The P-value is 0.40

16. Example: Are Astrologers’ Predictions Better Than Guessing?
Step 5: Conclusion
The P-value of 0.40 is not especially small
It does not provide strong evidence against H0: p = 1/3
There is not strong evidence that astrologers have special predictive powers

17. How Do We Interpret the P-value?
A significance test analyzes the strength of the evidence against the null hypothesis
We start by presuming that H0 is true
The burden of proof is on Ha

18. How Do We Interpret the P-value?
The approach used in hypotheses testing is called a proof by contradiction
To convince ourselves that Ha is true, we must show that data contradict H0
If the P-value is small, the data contradict H0 and support Ha

19. Two-Sided Significance Tests
A two-sided alternative hypothesis has the form Ha: p ≠ p0
The P-value is the two-tail probability under the standard normal curve
We calculate this by finding the tail probability in a single tail and then doubling it

20. Example: Dr Dog: Can Dogs Detect Cancer by Smell?
Study: investigate whether dogs can be trained to distinguish a patient with bladder cancer by smelling compounds released in the patient’s urine

21. Example: Dr Dog: Can Dogs Detect Cancer by Smell?
Experiment:
Each of 6 dogs was tested with 9 trials
In each trial, one urine sample from a bladder cancer patient was randomly place among 6 control urine samples

22. Example: Dr Dog: Can Dogs Detect Cancer by Smell?
Results:
In a total of 54 trials with the six dogs, the dogs made the correct selection 22 times (a success rate of 0.407)

23. Example: Dr Dog: Can Dogs Detect Cancer by Smell?
Does this study provide strong evidence that the dogs’ predictions were better or worse than with random guessing?

24. Example: Dr Dog: Can Dogs Detect Cancer by Smell?
Step 1: Check the sample size requirement:
Is the sample size sufficiently large to use the hypothesis test for a population proportion?
Is np0 >15 and n(1-p0) >15?
54(1/7) = 7.7 and 54(6/7) = 46.3
The first, np0 is not large enough
We will see that the two-sided test is robust when this assumption is not satisfied

25. Example: Dr Dog: Can Dogs Detect Cancer by Smell?
Step 2: Hypotheses
H0: p = 1/7
Ha: p ≠ 1/7

26. Example: Dr Dog: Can Dogs Detect Cancer by Smell?
Step 3: Test Statistic

27. Example: Dr Dog: Can Dogs Detect Cancer by Smell?
Step 4: P-value

28. Example: Dr Dog: Can Dogs Detect Cancer by Smell?
Step 5: Conclusion
Since the P-value is very small and the sample proportion is greater than 1/7, the evidence strongly suggests that the dogs’ selections are better than random guessing

29. Summary of P-values for Different Alternative Hypotheses
Alternative Hypothesis
P-value
Ha: p > p0
Right-tail probability
Ha: p < p0
Left-tail probability
Ha: p ≠ p0
Two-tail probability

30. The Significance Level Tells Us How Strong the Evidence Must Be
Sometimes we need to make a decision about whether the data provide sufficient evidence to reject H0
Before seeing the data, we decide how small the P-value would need to be to reject H0
This cutoff point is called the significance level

31. The Significance Level Tells Us How Strong the Evidence Must Be

32. Significance Level
The significance level is a number such that we reject H0 if the P-value is less than or equal to that number
In practice, the most common significance level is 0.05
When we reject H0 we say the results are statistically significant

33. Possible Decisions in a Test with Significance Level = 0.05
P-value:
Decision about H0:
≤ 0.05
Reject H0
> 0.05
Fail to reject H0

34. Report the P-value
Learning the actual P-value is more informative than learning only whether the test is “statistically significant at the 0.05 level”
The P-values of 0.01 and are both statistically significant in this sense, but the first P-value provides much stronger evidence against H0 than the second

35. “Do Not Reject H0” Is Not the Same as Saying “Accept H0”
Analogy: Legal trial
Null Hypothesis: Defendant is Innocent
Alternative Hypothesis: Defendant is Guilty
If the jury acquits the defendant, this does not mean that it accepts the defendant’s claim of innocence
Innocence is plausible, because guilt has not been established beyond a reasonable doubt

36. One-Sided vs Two-Sided Tests
Things to consider in deciding on the alternative hypothesis:
The context of the real problem
In most research articles, significance tests use two-sided P-values
Confidence intervals are two-sided

37. The Binomial Test for Small Samples
The test about a proportion assumes normal sampling distributions for and the z-test statistic.
It is a large-sample test the requires that the expected numbers of successes and failures be at least 15. In practice, the large-sample z test still performs quite well in two-sided alternatives even for small samples.
Warning: For one-sided tests, when p0 differs from 0.50, the large-sample test does not work well for small samples

38. Significance Tests about Means
Section 8.3
Significance Tests about Means

39. What Are the Steps of a Significance Test about a Population Mean?
Step 1: Assumptions
The variable is quantitative
The data are obtained using randomization
The population distribution is approximately normal. This is most crucial when n is small and Ha is one-sided.

40. What Are the Steps of a Significance Test about a Population Mean?
Step 2: Hypotheses:
The null hypothesis has the form:
H0: µ = µ0
The alternative hypothesis has the form:
Ha: µ > µ0 (one-sided test) or
Ha: µ < µ0 (one-sided test) or
Ha: µ ≠ µ0 (two-sided test)

41. What Are the Steps of a Significance Test about a Population Mean?
Step 3: Test Statistic
The test statistic measures how far the sample mean falls from the null hypothesis value µ0 relative to what we’d expect if H0 were true
The test statistic is:

42. What Are the Steps of a Significance Test about a Population Mean?
Step 4: P-value
The P-value summarizes the evidence
It describes how unusual the data would be if H0 were true

43. What Are the Steps of a Significance Test about a Population Mean?
Step 5: Conclusion
We summarize the test by reporting and interpreting the P-value

44. Summary of P-values for Different Alternative Hypotheses
Alternative Hypothesis
P-value
Ha: µ > µ0
Right-tail probability
Ha: µ < µ0
Left-tail probability
Ha: µ ≠ µ0
Two-tail probability

45. Example: Mean Weight Change in Anorexic Girls
A study compared different psychological therapies for teenage girls suffering from anorexia
The variable of interest was each girl’s weight change: ‘weight at the end of the study’ – ‘weight at the beginning of the study’

46. Example: Mean Weight Change in Anorexic Girls
One of the therapies was cognitive therapy
In this study, 29 girls received the therapeutic treatment
The weight changes for the 29 girls had a sample mean of 3.00 pounds and standard deviation of 7.32 pounds

47. Example: Mean Weight Change in Anorexic Girls

48. Example: Mean Weight Change in Anorexic Girls
How can we frame this investigation in the context of a significance test that can detect a positive or negative effect of the therapy?
Null hypothesis: “no effect”
Alternative hypothesis: therapy has “some effect”

49. Example: Mean Weight Change in Anorexic Girls
Step 1: Assumptions
The variable (weight change) is quantitative
The subjects were a convenience sample, rather than a random sample. The question is whether these girls are a good representation of all girls with anorexia.
The population distribution is approximately normal

50. Example: Mean Weight Change in Anorexic Girls
Step 2: Hypotheses
H0: µ = 0
Ha: µ ≠ 0

51. Example: Mean Weight Change in Anorexic Girls
Step 3: Test Statistic

52. Example: Mean Weight Change in Anorexic Girls
Step 4: P-value
Minitab Output
Test of mu = 0 vs not = 0
Variable N Mean StDev SE Mean wt_chg CI
95% CI T P
( , )

53. Example: Mean Weight Change in Anorexic Girls
Step 5: Conclusion
The small P-value of provides considerable evidence against the null hypothesis (the hypothesis that the therapy had no effect)

54. Example: Mean Weight Change in Anorexic Girls
“The diet had a statistically significant positive effect on weight (mean change = 3 pounds, n = 29, t = 2.21, P-value = 0.04)”
The effect, however, may be small in practical terms
95% CI for µ: (0.2, 5.8) pounds

55. Results of Two-Sided Tests and Results of Confidence Intervals Agree
Conclusions about means using two-sided significance tests are consistent with conclusions using confidence intervals
If P-value ≤ 0.05 in a two-sided test, a 95% confidence interval does not contain the H0 value
If P-value > 0.05 in a two-sided test, a 95% confidence interval does contain the H0 value

56. What If the Population Does Not Satisfy the Normality Assumption
For large samples (roughly about 30 or more) this assumption is usually not important
The sampling distribution of x is approximately normal regardless of the population distribution

57. What If the Population Does Not Satisfy the Normality Assumption
In the case of small samples, we cannot assume that the sampling distribution of x is approximately normal
Two-sided inferences using the t distribution are robust against violations of the normal population assumption
They still usually work well if the actual population distribution is not normal

58. Regardless of Robustness, Look at the Data
Whether n is small or large, you should look at the data to check for severe skew or for severe outliers
In these cases, the sample mean could be a misleading measure