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Check THIS website for an article about some of our superstars of statistics.
The applet on this webpage should be enlightening regarding Type I and Type II error. Scroll down the linked page a bunch until you find the yellow box entitled Statistical Errors Applet. Read the information in the little yellow box and click on the link where it says Applet 1. Statistical Errors near the little graph to display the applet. Move the box in the scroll bar at the top to change the value of alpha (the probability of a Type I error). In real life you can change this value, but pick your alpha before you collect your data! Otherwise, you might be accused of manipulating the data and giving statisticiansa bad name.
Now, move the middle of a NEW and IMPROVED distribution by sliding the box in the scroll bar at the bottom of the window. See what the yellow region looks like when you overlap the distributions. The yellow area represents the probability of a Type II error.
So, what effect does changing alpha have on the probabilty of a Type II error? When is beta maximized? When is it minimized?
TYPE I and TYPE II errors
PLEASE read the section in the book regarding these topics.
You will only be able to calculate a POWER or a BETA (the probability of a Type II error) when some NEW mean is introduced. The power of the test is the probability that the test will be able to distinguish between your original hypothesized mu and the newly proposed mu. The probability of an error is BETA.
To calculate BETA:
Find the boundaries of the FTR region for your original hypothesis. Find the probability that x-bar would fall between that lower bound and upper bound GIVEN the NEW mu and standard error of the mean. In calculator language [that you would NEVER write on a test] it would be normcdf(LB, UB, NEWmu, sigma of x-bar).
HW due Wednesday: 11.36-11.40. Your test is Thursday.
You should have worked problems 11.5, .6, .27, .28, .29, .30, .49, .50, & .51 by Tuesday. Your test is Thursday.
The excerpt in class today was from The Lady Tasting Tea by David Salsburg.
HW due Thursday: 11.3, 11.4, 11.6
Please note that (1) null hypotheses ALWAYS have an "equals" concept
(2) null and alternative hypotheses do NOT include statistics.
In inference testing, the results of our sample may make us reject the null hypothesis if they are so unlikely that they would be unbelievably unlikely due to randomness.
Please read through the top of page 693 AND register for the AP exam.
Monday, February 23, 2009
Wednesday, February 04, 2009
Chapter 10 Confidence Intervals
Your test on Chapter 10 will be Tuesday, February 17.
HW due Friday: Either problem 10.53 worked out in detail showing all work or problems 10.54 and 10.58 worked cout completely.
Have you hugged your study guide lately????
HW due Thursday: Problems 10.38 and .44. 1st and 2nd periods, please bring all HW from this week on Thursday.
Today we computed paired t confidence intervals for the difference in grip strength between right and left hands.
You can find the t* value for any number of df by using the calculator;
STAT TESTS T-INT Stats x-bar = 0, sx = sqrt df+1, n = df+1, conf level = whatever you need, like .95.
ALL students should have finished 10.28, 10.30, 10.31, and 10.31 PLUS the summary of the cautions. Have these with you on Wednesday.
HW for 1st and 2nd periods: Summarize the cautions of section 10.1 (pages 635-637) in your own words and work problems 10.28 and 10.30.
Periods 6 & 7: Work problems 10.7-10.10 PLUS summarize the cautions above.
The question was raised: Why do we use 2 sometimes and 1.96 other times for Z*? As you probably recall, approximately 95% of the data in a Normal distribution will fall within about 2 standard deviations of the mean, but that was just an estimate. The more precise number of standard deviations that form the 95% boundaries is 1.96. Use that whenever we are using Z procedures UNLESS we are just looking for a quick and dirty estimate. but NOT when we are constructing confidence intervals.
When do we use sx and when do we use sigmax? Sigma represents the population standard deviation, a number we rarely know. On the other hand, sx represents our sample standard deviation. When we do not know the population standard deviation we will use t procedures instead of z procedures.
And, of course, we divide by sqrt of n to convert these standard deviations into standard errors of x-bar.
Some web-based applets for Confidence Intervals: Rice Univ Freeman
HW due Friday--
1st and 2nd per: Problems 10.7-10.10 from the text. You should have already worked the problems from the REVIEW III on pages 610 and 611.
6th and 7th per: "Review III" questions following Chapter 9 on pages 610-611 in the text AND print one page from a confidence interval applet from the web and be able to explain it.
Key concepts from today: Approximately 95% of sample averages will fall within about 2 std dev/Sqrt(n) of the population mean. If we don't know what the population mean is, we might reason that our point estimate (x-bar) is a pretty good guess, and that 95% of the time, our sample averages will fall within 2 std dev/sqrt(n) of the true mean. Then the interval (x-bar minus 2*std dev/sqrt(n), x-bar plus 2*std dev/sqrt(n)) is our confidence interval or reasonable guess at the value of the population mean. About 95% of these intervals will capture the true mean. The distance from the mean to the upper bound (or event the lower bound) is the margin of error.
These are NOT true: 95% of the time this interval contains the mean. 95% of population means fall inside this interval. 95% of the time the mean falls between lower bound and upper bound. NONE of these are true, so DO NOT write these as interpretations of the confidence intervals.
Instead, we are 95% confident that the mean falls between the lower bound and the upper bound.
If you did not work problems from the review following Chapter 9, now is the time!!!!
HW due Friday: Either problem 10.53 worked out in detail showing all work or problems 10.54 and 10.58 worked cout completely.
Have you hugged your study guide lately????
HW due Thursday: Problems 10.38 and .44. 1st and 2nd periods, please bring all HW from this week on Thursday.
Today we computed paired t confidence intervals for the difference in grip strength between right and left hands.
You can find the t* value for any number of df by using the calculator;
STAT TESTS T-INT Stats x-bar = 0, sx = sqrt df+1, n = df+1, conf level = whatever you need, like .95.
ALL students should have finished 10.28, 10.30, 10.31, and 10.31 PLUS the summary of the cautions. Have these with you on Wednesday.
HW for 1st and 2nd periods: Summarize the cautions of section 10.1 (pages 635-637) in your own words and work problems 10.28 and 10.30.
Periods 6 & 7: Work problems 10.7-10.10 PLUS summarize the cautions above.
The question was raised: Why do we use 2 sometimes and 1.96 other times for Z*? As you probably recall, approximately 95% of the data in a Normal distribution will fall within about 2 standard deviations of the mean, but that was just an estimate. The more precise number of standard deviations that form the 95% boundaries is 1.96. Use that whenever we are using Z procedures UNLESS we are just looking for a quick and dirty estimate. but NOT when we are constructing confidence intervals.
When do we use sx and when do we use sigmax? Sigma represents the population standard deviation, a number we rarely know. On the other hand, sx represents our sample standard deviation. When we do not know the population standard deviation we will use t procedures instead of z procedures.
And, of course, we divide by sqrt of n to convert these standard deviations into standard errors of x-bar.
Some web-based applets for Confidence Intervals: Rice Univ Freeman
HW due Friday--
1st and 2nd per: Problems 10.7-10.10 from the text. You should have already worked the problems from the REVIEW III on pages 610 and 611.
6th and 7th per: "Review III" questions following Chapter 9 on pages 610-611 in the text AND print one page from a confidence interval applet from the web and be able to explain it.
Key concepts from today: Approximately 95% of sample averages will fall within about 2 std dev/Sqrt(n) of the population mean. If we don't know what the population mean is, we might reason that our point estimate (x-bar) is a pretty good guess, and that 95% of the time, our sample averages will fall within 2 std dev/sqrt(n) of the true mean. Then the interval (x-bar minus 2*std dev/sqrt(n), x-bar plus 2*std dev/sqrt(n)) is our confidence interval or reasonable guess at the value of the population mean. About 95% of these intervals will capture the true mean. The distance from the mean to the upper bound (or event the lower bound) is the margin of error.
These are NOT true: 95% of the time this interval contains the mean. 95% of population means fall inside this interval. 95% of the time the mean falls between lower bound and upper bound. NONE of these are true, so DO NOT write these as interpretations of the confidence intervals.
Instead, we are 95% confident that the mean falls between the lower bound and the upper bound.
If you did not work problems from the review following Chapter 9, now is the time!!!!
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