1 Chapter 23 Inferences About Means

2 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.

3 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,.

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

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

6 Solution Use the sample standard deviation, s and the standard error of

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

8 Solution The distribution of the standardized sample mean does follow a Student’s t-model with df = n – 1.

9 Inference for Do NOT use Table Z! Use Table T instead! Table Z

10

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

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

13 10% Condition The sample is no more than 10% of the population. Not as critical for means as it is for proportions.

14 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?

15 Confidence Interval for

16 Table T Confidence Levels 80% 90% 95% 98% 99% df n–1

17 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,.

18 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

19 Summary of Data n = 25 = beats s = beats = beats

20 Conditions Randomization condition: met. 10% condition: met. Nearly normal condition: met.

21 Heart rate

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

23 Confidence Interval for

24 Table T Confidence Levels 80% 90% 95% 98% 99% df

25 Confidence Interval for

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

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

28 JMP: Analyze – Distribution Mean74.16 Std Dev5.375 Std Err Mean1.075 Upper 95% Mean76.38 Lower 95% Mean71.94 N25

29 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.

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

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

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

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

34 Summary of Data n = 25 = beats s = beats = beats

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

36 Table T One tail probability P-value df The P-value is less than

37 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.

38 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.

39 Alternatives

40 JMP: Analyze – Distribution Test Mean t-test Hypothesized value70Test statistic 3.87 Actual Estimate74.16Prob > |t| df24Prob > t Std Dev5.375Prob < t0.9996

41 Another Example Is the population mean octane rating 90 or is it something different?

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

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

44 Octane Rating

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

46 Summary of Data n = 40 = s = =

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

48 Table T Two tail probability P-value df The P-value is less than 0.01.

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

50 Step 5: State your conclusion within the context of the problem. The population mean octane rating is not 90 but something different. Test of Hypothesis for

51 Test of Hypothesis for

52 Interpretation We are 95% confident that the population mean octane rating is between and 91.44

53 Interpretation This confidence interval agrees with the test of hypothesis. 90 is not in the interval and so must be rejected as a value for the population mean.