1. Inference for Who? Young adults. What? Heart rate (beats per minute).
When?
Where? In a physiology lab.
How? Take pulse at wrist for one minute.
Why? Part of an evaluation of general health.

2. Inference for What is the mean heart rate for all young adults?
Use the sample mean heart rate, , to make inferences about the population mean heart rate, .

3. Inference for Sampling distribution of Shape: Approximately normal
Center: Mean,
Spread: Standard Deviation,

4. Problem The population standard deviation, is unknown.
Therefore, is unknown as well.

5. Solution
Use the sample standard deviation, s to get the standard error of

6. Problem The distribution of the standardized sample mean
does not follow a normal model.

7. Solution The distribution of the standardized sample mean
does follow a Student’s t-model with df = n – 1.
A little history about Student’s t model. This model was developed by an individual working at the Guinness Brewery in Dublin Ireland.

8. Guinness is one of the oldest breweries in the world and has been brewing beer at the St. James’s Gate Brewery in Dublin since In the 1920 a brewer, who was also a statistician, named William Sealy Gosset was working on data on barley. He discovered that his inferences were not as good as was predicted by theory and he set about to find out why. The reason, Table Z was not appropriate when using the sample standard deviation. Gosset went on to develop Table T to overcome this problem. Because Guinness had a policy against publishing any results of research coming out of the brewery, Gosset had to publish his results under a pseudonym – A. Student.

9. Later, William Gosset went on to be the Master Brewer for Guinness.

10. Inference for
Do NOT use Table Z!
Use Table T instead!
Table Z

12. Conditions Randomization condition. 10% condition.
Nearly normal condition.

13. Randomization Condition
Data arise from a random sample from some population.
Data arise from a randomized experiment.

14. 10% Condition The sample is no more than 10% of the population.
Not as critical for means as it is for proportions.

15. Nearly Normal Condition
The data come from a population whose shape is unimodal and symmetric.
Look at the distribution of the sample.
Could the sample have come from a normal model?

16. Confidence Interval for

17. Table T df 1 2 3 4
n–1
Confidence Levels 80% % % % %

18. Inference for What is the mean heart rate for all young adults?
Use the sample mean heart rate, , to make inferences about the population mean heart rate, .

19. Sample Data Random sample of n = 25 young adults.
Heart rate – beats per minute
70, 74, 75, 78, 74, 64, 70, 78, 81, 73
82, 75, 71, 79, 73, 79, 85, 79, 71, 65
70, 69, 76, 77, 66

20. Summary of Data
n = 25
= beats
s = beats
= beats

21. Conditions Randomization condition: 10% condition:
Met because we have a random sample of 25 young adults.
10% condition:
Met because 25 is less than 10% of all young adults that could have been sampled.

22. Conditions Nearly Normal Condition
Could the sample have come from a population described by a normal model?

23. Heart rate

24. Nearly Normal Condition
Normal quantile plot – data follows straight line for a normal model.
Box plot – symmetric.
Histogram – unimodal and symmetric.

25. Confidence Interval for

26. Table T df 1 2 3 4 24
2.064
Confidence Levels 80% % % % %

27. Confidence Interval for

28. 71.94 beats/min and 76.38 beats/min
Interpretation
We are 95% confident that the population mean heart rate of young adults is between
71.94 beats/min and beats/min

29. Interpretation Plausible values for the population mean.
95% of intervals produced using random samples will contain the population mean.

30. JMP:Analyze – Distribution
Mean
74.16
Std Dev
5.375
Std Err Mean
1.075
Upper 95% Mean
76.38
Lower 95% Mean
71.94 N 25

31. Test of Hypothesis for
Could the population mean heart rate of young adults be 70 beats per minute or is it something higher?

32. Test of Hypothesis for
Step 1: State your null and alternative hypotheses.

33. Test of Hypothesis for Step 2: Check conditions.
Randomization condition, met.
10% condition, met.
Nearly normal condition, met.

34. Test of Hypothesis for
Step 3: Calculate the test statistic and convert to a P-value.

35. Summary of Data
n = 25
= beats
s = beats
= beats

36. Value of Test Statistic
Use Table T to find the P-value.

37. Table T The P-value is less than 0.005. 3.87
One tail probability P-value df 1 2 3 4 24
3.87
The P-value is less than

38. Test of Hypothesis for Step 4: Use the P-value to reach a decision.
The P-value is very small, therefore we should reject the null hypothesis.

39. Test of Hypothesis for
Step 5: State your conclusion within the context of the problem.
The mean heart rate of all young adults is more than 70 beats per minute.

40. Alternatives

41. JMP:Analyze – Distribution
Test Mean
t-test
Hypothesized value 70
Test statistic
3.87
Actual Estimate
74.16
Prob > |t|
0.0007 df 24
Prob > t
0.0004
Std Dev
5.375
Prob < t
0.9996

42. Example What is the mean alcohol content of beer?
A random sample of 10 beers is taken and the alcohol content (%) is measured.

43. Sample Data – Alcohol (%)
Molson Canadian
5.19
Heineken Dark
5.17
Michelob Dark
4.76
O’Keefe Canadian
4.96
Big Barrel Lager
4.32
Olympia Lager
4.78
Hamm’s
4.53
Miller Draft
4.85
Tsingtao
4.79
Guinness Stout
4.27

44. Test of Hypothesis for
Step 1: State your null and alternative hypotheses.

45. Test of Hypothesis for Step 2: Check conditions.
Randomization condition, met.
10% condition, met.
Nearly normal condition, met.

46. Test of Hypothesis for
Step 3: Calculate the test statistic and convert to a P-value.

47. Test of Hypothesis for
Test statistic,

48. Table T The P-value is between 0.02 and 0.05.
Two tail probability P-value 0.02 df 1 2 3 4 9
The P-value is between 0.02 and 0.05.

49. Test of Hypothesis for Step 4: Use the P-value to reach a decision.
The P-value is smaller than 0.05, therefore we should reject the null hypothesis.

50. Test of Hypothesis for
Step 5: State your conclusion within the context of the problem.
The population mean alcohol content of beer is not 5%.

51. JMP Output

52. Confidence Interval for

53. Interpretation
We are 95% confident that the population mean alcohol content of beer is between 4.537% and 4.987%.

54. Interpretation
The population mean alcohol content of beer could be any value between 4.537% and 4.987%.
If we repeat the procedure that produces a confidence interval, 95% of intervals produced will capture the population mean.