Unformatted text preview: 2/16/2011 Lecture 8: February 16, 2011
• •
• • • Readings for today:
– Two‐Sample Hypothesis Tests: Paired Samples, pages 404‐409
– This is the last topic for Exam 1.
Readings for next class:
– Comparing Two Variances (pp. 417‐424)
Homework 6:
– Opened at 11:00 p.m. February 14
– Page 401: Problems 10.4 and 10.6 – Two‐sample test, assume equal variances
– Closes Thursday, February 17, at 6:00 p.m.
– Unlimited tries until it closes! Feedback!
Connect Quiz No. 1:
– Will probably open late Thursday – Will probably close next Wednesday at 6:00 p.m.
– You can expect 15‐20 questions.
– You will have 2 two‐hours sessions to complete the quiz.
Exam No. 1:
– Friday, February 25
– Begins at 3:30 p.m.
– Seating arrangements will be announced next week. Installing Minitab on a MAC
• Go to the course web site.
• Go to the bottom menu
– Installing Minitab on a MAC •
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•
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• You will need to install Windows 7.
Don’t let this scare you!
You get a free copy.
Read the installation directions.
Don’t try installing tonight. Hyperlinks will not work until tomorrow. Discussion Session 5: Friday, February 18
• First item of business: Any issues with Worked‐Out Problems: Problem 10.3 and Problem 10.4, page 401.
• Second item of business: Go to the “Discussion Data Sets “ menu on my web site. Download the data set titled “Discussion 5 Data, Pulse Readings.” As background, there are two sets of data in this spreadsheet. Sheet 1 consists of 46 individuals who initially rest and then run in place. Sheet 2 consists of 63 individuals who rest and then rest again. Notice that each sheet amounts to a “before” and “after” activity. • Third item of business: You will be implementing a statistical technique that compares “before” and “after” groups. The question is: Would you expect a difference in pulse rate before and after each activity? The technique is a paired‐sample t‐test. We will do this technique today in class. On Friday, you will be doing the test by hand using the data on Sheet 1. Your TA will supply you with calculations for key summary statistics. Next, you will be using Minitab on the data in Sheet 2. 1 2/16/2011 Comparing Two Means: Paired Samples
(Let’s promote this with an example).
• Ten cars are selected at random.
• Each is driven both with and without a gasoline additive that is supposed to increase mileage.
• Notice: The wording suggests an individual pairing before any kind of summary measure is calculated.
• Test the null hypothesis that the mean gas mileage with the additive is the same as without the additive versus the alternative hypothesis that the mean gas mileage with the additive is greater than without the additive.
• Test at the five‐percent level. First, the data (in miles per gallon)
• Car No. With additive (w/A)
Without Additive (wo/A)
•
1 25.7 24.9
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2 20.0 18.8
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3 28.4 27.7
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4 13.7 13.0
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5 18.8 17.8
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6 12.5 11.3
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7 28.4 27.8
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8 8.1 8.2
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9 23.1 23.1
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10 10.4 9.9 The Sampling Distribution for Paired Differences
• If we could calculate the paired difference for every conceivable automobile
– and then compute the population mean of all of the paired differences,
– this mean would be denoted as µd.
• Likewise, if we took repeated samples of 10 cars,
– calculated differences with and without the additive, – and then calculated a sample mean for each set of 10 differences,
– we would have a sampling distribution of sample mean differences based on repeated samples of size 10. • Let the sample mean of the differences be referred to as .
d
• Here’s a picture of the sampling distribution. 2 2/16/2011 Sampling Distribution of d d
d Setting Up the Hypotheses
• Notation:
– w/A: with additive
– wo/A: without additive
– di = paired difference for pair “i” • The symbol used in both hypotheses is µd.
• H0: µd = 0.
(or, in words, there is no difference due to the additive) • H1: µd > 0
(or, in words, the additive results in an increase in mpg) Sampling Distribution of d
(under the assumption that H₀ is true) d
d 0 3 2/16/2011 Comparing Two Means: Paired Samples
(Here are the paired‐difference calculations. ) Pairing Introduces Valuable Information
• If you treated the two columns of information as independent samples – and simply did the “vertical” calculations – you’d be ignoring valuable information.
• This valuable information is contained in the “horizontal” calculations associated with differencing each pair. Let’s do some summary calculations on the paired differences.
• What should these be?
• How about the old stand‐bys:
– Sample mean
– Sample standard deviation
– Standard error of the mean
– While we’re at it:
• Test statistic
• Standardized test statistic 4 2/16/2011 Comparing Two Means: Paired Samples
The following are summary statistics on the individual di s
(See page 404 for the formulas below)
n d= sd = d i i=1 n d d n n1 i=1 t= 2 i dμ d
sd
n Do you recall how to calculate the standard deviation sd ? Comparing Two Means: Paired Samples
The following are summary statistics on the individual di s
(See page 404 for the formulas below)
n d= sd = d n i=1 t= i i=1 n d d 2 i n1
dμ d
sd
n 5 2/16/2011 Remember!
• Any time we develop a test statistic using t
• We must have an appropriate calculation for degrees of freedom.
• Look again at the standardized test statistic.
• This test has been converted to a test of one
mean.
• Specifically, it is a test of the mean difference.
• For a test of one mean, df = n‐1.
• For our example, df = 10‐1 = 9. Hypothesis Test Steps : Gas Additive vs No Gas Additive: Paired Samples
(1) Set up H0 and HA.
(2) Establish the decision rule.
(a) Select .
(b) Critical value t .
(c) “Fail to Reject H0” region.
“Reject H0 ” region.
(3) (a) Test statistic
(b) Standardized test statistic
(4) Decision items:
(a) Place t on decision‐rule line.
(b) Make a decision. Mathematical reason.
(5) Evidence to support what the
alternative (or research) hypothesis purports to show? Hypothesis Test Steps : Gas Additive vs No Gas Additive: Paired Samples
(1) Set up H0 and HA.
(2) Establish the decision rule.
(a) Select .
(b) Critical value t .
(c) “Fail to Reject H0” region.
“Reject H0 ” region.
(3) (a) Test statistic
(b) Standardized test statistic
(4) Decision items:
(a) Place t on decision‐rule line.
(b) Make a decision. Mathematical reason.
(5) Evidence to support what the
alternative (or research) hypothesis purports to show? 6 ...
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