1. Lecture #18 Thursday, October 20, 2016 Textbook: Sections 12.1 to 12.4
Statistics 200
Lecture #18 Thursday, October 20, 2016
Textbook: Sections 12.1 to 12.4
Objectives:
• Formulate null and alternative hypotheses correctly based on the context of a testing situation.
• Distinguish between a one-sided and two-sided alternative hypothesis.
• Calculate a test statistic using standardization.
• Find a p-value based on a test statistic along with the direction of the alternative hypothesis.

2. Question from Midterm #2
Correct Answer: A

3. We have begun a strong focus on Inference
Means
Proportions
One population mean
One population proportion
Two population proportions
Difference between Means
Mean difference
This week

4. Now that we know how the sample proportion behaves, we can compare a specific sample estimate to this distribution.
Key to hypothesis testing: we state a null, get a distribution like this based on that null, and then see how unusual our sample estimate is in comparison.

5. Statistical Hypotheses
Null Hypothesis, H0:
Nothing happening
No change / difference
Alternative Hypothesis, Ha:
Something is happening
There is a change / difference

6. Clicker Quiz # 11: Quoted Number: (3 in 10) is:
Major Study found: 3 in 10 adults acknowledged that they have nodded off while driving within the past year.
Clicker Quiz # 11:
Quoted Number: (3 in 10) is:
A. odds B. relative risk
C. increased risk D. individual risk

7. Drowsy Driving Example: Hypothesis Test
When considering all drivers about three in 10 indicate that they have driven while drowsy. Research Question: Does recent evidence suggest that for young drivers ages 16–24, this risk is actually higher than 0.3? State hypotheses using appropriate language. In the population: H0: drivers 16–24 years old have the ______ risk as 0.3. Ha: drivers 16–24 years old have a ________ risk than 0.3.
same
higher
This is a one-sided alternative.

8. Drowsy Driving Example cont’d
Rewrite the hypotheses using the appropriate Parameters: Research Question: Recent evidence suggests that with drivers ages 16–24, this rate is actually higher H0: p = 0.3 Ha: p > 0.3 (Here, p represents the population proportion of drivers aged 16–24 who have nodded off while driving in the past year.)
Important! Hypotheses are always about population parameters, never sample statistics.

9. Question from Midterm #2
Correct Answer: A

10. Hypothesis Test: Steps
Data Summary:
test statistic
Find the
p-value
Make a
decision
State a
conclusion
State
Hypotheses

11. Calculate the sample statistic
A sample of 300 drivers from the 16–24 age group found 105 who say that they have driven while drowsy in the last year.
This is not (yet) the test statistic. We need to standardize it by subtracting the mean of p-hat and dividing by the SD of p-hat under the null.

12. Calculate the test statistic
This is not (yet) the test statistic. We need to standardize it by subtracting the mean of p-hat and dividing by the SD of p-hat under the null.
H0: p = Ha: p > 0.30

13. General test statistic formula:
In our example, this formula leads to:

14. Question from Midterm #2
Correct Answer: B

15. Hypothesis Test: Steps
We have a test statistic equal to
Data Summary:
test statistic
Find the
p-value
Make a
decision
State a
conclusion
State
Hypotheses
Also, the alternative is
Ha: p > 0.30.
We can use this info to find the p-value.

16. p-value definition We have a test statistic equal to 1.890.
The p-value is the probability, if H0 is true, that our experiment would give a test statistic at least as extreme as the test statistic we observed.
Also, the alternative is
Ha: p > 0.30.
“At least as extreme” means in the direction determined by the alternative hypothesis.
In this case, the p-value is P(Z≥1.890).
Therefore, the p-value is

17. Recall this example: Are women more likely to have dogs?
Has Dog
No Dog
Total
Female 89
56.7% 68
43.3%
157
Male 66
50.8% 64
49.2%
130
155
132
287
Your class data

18. Recall this example: Are women more likely to have dogs?
Has Dog
No Dog
Total
Female 89
56.7% 68
43.3%
157
Male 66
50.8% 64
49.2%
130
155
132
287
Let’s reframe this problem: Examine the difference between two independent proportions, that is, pf–pm.
Is it zero? Let’s run a statistical hypothesis test.

19. Recall this example: Are women more likely to have dogs?
Has Dog
No Dog
Total
Female 89
56.7% 68
43.3%
157
Male 66
50.8% 64
49.2%
130
155
132
287
This is a two-sided alternative.
H0: pf–pm = 0
Ha: pf–pm ≠ 0
Hypotheses:
In this dataset,

20. The sampling distribution of
As long as both p-hat1 and p-hat2 are approximately normal…
...and the two samples are independent...
Then the sampling distribution is approximately normal with mean p1–p2 and standard deviation

21. Recall the general test statistic formula:
In our example, the parameter is pf–pm.
Therefore:
• The sample estimate is
• The mean under H0 is 0
• The std dev. under H0 is
Notice: Same value of p-hat in both fractions!
That value is the combined sample proportion:

22. Recall the general test statistic formula:
In our example, the parameter is pf–pm.
Therefore:
• The sample estimate is
• The mean under H0 is 0
• The std dev. under H0 is
Conclusion: The test statistic is

23. p-value definition We have a test statistic equal to 1.00.
The p-value is the probability, if H0 is true, that our experiment would give a test statistic at least as extreme as the test statistic we observed.
Also, the alternative is
Ha: pf–pm ≠ 0.
“At least as extreme” means in the direction determined by the alternative hypothesis.
In this case, the p-value is P(Z≥1.00 or Z≤–1.00).
Therefore, the p-value is

24. Chi-square statistic: 1.003
Recall result from Lecture 08 (Sept. 15): Are women more likely to have dogs?
Has Dog
No Dog
Total
Female 89
56.7% 68
43.3%
157
Male 66
50.8% 64
49.2%
130
155
132
287
Chi-square statistic: 1.003
P-value: 0.317
Note: There was a mistake in the Sept. 15 calculation!

25. Question from Midterm #2
Correct Answer: C

26. If you understand today’s lecture…
12.25, 12.27, 12.41(b-d), 12.47, 12.55, 12.60, 12.63, 12.65, 12.66
Objectives:
• Formulate null and alternative hypotheses correctly based on the context of a testing situation.
• Distinguish between a one-sided and two-sided alternative hypothesis.
• Calculate a test statistic using standardization.
• Find a p-value based on a test statistic along with the direction of the alternative hypothesis.