Apr 10th - Stony Brook University
Apr 11th
Chapters 6&7 (continued):
Approach 1 – Pivotal Quantity Method
1. Inference on one population proportion p for large sample test: summary:
Same: [pic] [pic] [pic]
[pic] [pic] [pic]
Same( test statistic [pic]when Ho is true
At the significance level[pic], we reject Ho in favor of Ha if:
[pic] [pic] [pic]
[pic] [pic] [pic]
Definition: P-value (observed significance level) is the probability to observe the test statistic value or values more extreme than the test statistic value (here being Zo), given that Ho is true.
[pic][pic] [pic]
* Make conclusion using the P-value.
At the significance level[pic], we reject Ho in favor of Ha if the P-value is less than[pic].
* P-value is more informative.
2、Inference on one population mean[pic]for large sample test: Summary:
Same: [pic] [pic] [pic]
[pic] [pic] [pic]
Same( Test statistic:
Pivotal quantity: [pic] when [pic]is unknown.
[pic] When [pic]is known.
(*Z follows exactly N(0,1) distribution when the population is normal)
Test statistic: [pic]or [pic]
At the significance level[pic], we reject Ho if:
[pic] [pic] [pic]
P-values are the same as above (proportion)
Derivation for the 2-sided test: [pic]P (Type 1 error)=P (reject Ho|Ho)
[pic]
Hence [pic]
Therefore, we reject Ho at[pic]if: [pic]
Example 1 (Problem 6.2.10 from Page439)
At a class research project, Rosaura wants to see whether the stress of final exams elevates the blood pressures of freshmen women. When they are not under any untoward duress, healthy 18-year-old women have systolic blood pressures that average 120mmHg. If Rosaura finds that the average blood pressure for the 50 women in statistics 101 on the day of the final exam is 125.2 with sample standard deviation 13 mm Hg, what should she conclude? Set up and test an appropriate hypothesis.
Solution: This is test on one population mean for large sample.
N=50,[pic],[pic] [pic][pic]
[pic]
[pic]
Since Zo=2.8>1.645, we reject Ho at[pic]
P-value=0.0026, since P-value1.7959 hence we reject the null hypothesis.
Approach 2 -- Likelihood ratio test (LRT):
[pic]
Data:[pic]: i.i.d. follow p.d.f. f (x;[pic])
Let [pic]
Likelihood function: [pic]
[pic]
If Ho is true, then [pic] so we should reject Ho if [pic]
Use the definition of Type I error rate (significance level) to derive the threshold value c as usual. That is: Set [pic] and then solve for c
Example 3: Let [pic],[pic]is unknown. Please derive the LRT for[pic].
[pic] [pic]
[pic] [pic]
[pic]
[pic]
[pic] MLE’s for[pic]and[pic]in [pic]
For the numerator:
[pic] (*)
Then [pic]
Hence [pic]
Similarly, we can derive:
For the denominator:
[pic]
[pic]
Therefore, [pic]
At the significance level [pic], we reject the null hypothesis if [pic]
c is the value such that:
[pic]
[pic]
And, [pic]
[pic]
Hence the above formula becomes:
[pic] Given [pic]
Symmetric, therefore we reject Ho at significance level[pic] if:
[pic]
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