Mar 29th - Applied Mathematics & Statistics
Mar 29th
Chapters 6&7 (continued):
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]
P-value=[pic] P-value=[pic] P-value=[pic]
[pic] [pic] [pic]
Definition: P-value (observed significance level) is the probability to observe the test statistic value or more extreme values than the significance test statistic value—Zo, given that Ho is true (= a value that is at least as extreme as the test statistic value observed)
* 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: [pic]P (Type 1 error)=P (reject Ho|Ho)
[pic]
Hence [pic]
Therefore, we reject Ho at[pic]if: [pic]
Example 3a (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 with a standard deviation of 12mm Hg. 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, 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=3>1.645, we reject Ho at[pic]
P-value=0.013, since P-value ................
................
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