section 10.2 hypothesis testing for means (small samples) hawkes learning systems math courseware...
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Section 10.2
Hypothesis Testing for Means (Small Samples)
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2008 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
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What this lesson is about
• Learn to perform a hypothesis test• The previous lesson was only about how to set
up a hypothesis test.– Reading and understanding the real-life scenario.– Getting the right letter, μ or p.– Getting the right relational operators in the right
places: = and ≠, ≤ and >, ≥ and <.– Getting the right value of μ or p (and setting aside
the “noise” numbers in the problem statement.)
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Choice: Do a t Test or a z Test?
Small Samples: t Test• “Small” means “sample size
is n < 30.• There’s an assumption that
the population is normally distributed.
• If the population is not normally distributed, this method we use is NOT valid.
• Easy for today: everything we do is a t Test.
Large Samples: z Test• “Large” means “sample size
is n ≥ 30.• To be discussed in a later
lesson.• The Bluman book has
slightly different rules from the way this Hawkes book does it. Just be aware of that.
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Test Statistic for Small Samples, n < 30:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
with d.f. = n – 1
To determine if the test statistic calculated from the sample is statistically significant we will need to look at the critical value.
The critical values for n < 30 are found from the t-distribution.
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Find the critical value:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
Find the critical t-score for a right-tailed test that has 14 degrees of freedom at the 0.025 level of significance.
Solution:
d.f. = 14 and a = 0.025
t0.025 = 2.145
(Added info)• It’s in Table C, Critical Values
of t
Inputs: • Column for α (alpha)• Choose the right column for
one- or two-tailed• Row for d.f., degrees of
freedom (= sample size n, minus 1)
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Rejection Regions:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
Determined by two things:1. The type of hypothesis test.
2. The level of significance, a.
Finding a Rejection Region:
1. Look up the critical value, tc, to determine the cutoff for the rejection region.
2. If the test statistic you calculate from the sample data falls in the a area, then reject H0.
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Types of Hypothesis Tests:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
Alternative Hypothesis
< Value> Value≠ Value
Type of Test
Left-tailed testRight-tailed testTwo-tailed test
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Rejection Regions for Left-Tailed Tests, Ha contains <:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
Reject if t ≤ –t
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Rejection Regions for Right-Tailed Tests, Ha contains >:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
Reject if t ≥ t
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Rejection Regions for Two-Tailed Tests, Ha contains ≠:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
Reject if | t | ≥ t/2
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Steps for Hypothesis Testing:
1. State the null and alternative hypotheses.2. Set up the hypothesis test by choosing the
test statistic [that is, make a decision on whether it’s a t or z problem] and determining the values of the test statistic that would lead to rejecting the null hypothesis [the critical value(s)].
3. Gather data and calculate the necessary sample statistics [t = or z = ].
4. Draw a conclusion [Stating it two ways: reject/fail to reject, and also in plain English].
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
IMPORTANT !!!!
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Draw a conclusion:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
27 college students are asked how many parking tickets they get a semester to test the claim that the average student gets more than 9 tickets a semester. The sample mean for the number of tickets was found to be 9.8 with a standard deviation of 1.5. Use a = 0.10.
Solution:
n = 27, = 9, = 9.8, s = 1.5, d.f. = 26, a = 0.10
t0.10 =
Since t is greater than t a , we will reject the null hypothesis.
1.315
2.771
H0: μ ≤ 9 tickets Ha: μ > 9 tickets.
This is the CRITICAL VALUE. Either use table or invT(0.10,26). Draw a PICTURE, too. Mark 1.315and highlight the critical region.
This is the TEST STATISTIC.Mark 2.771 on your picture.
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Remarks about the parking ticket example
• There was a choice made to do a t Test because the sample size was < 30.
• There was an implicit assumption that the distribution of the count of parking tickets fits a normal distribution.
• It was a RIGHT-TAILED TEST because of the “>” in the alternative hypothesis.
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Remarks about the parking ticket example, continued
• Hypothesis tests are really essay questions. • The outline for the essay is the four-step
procedure described in the earlier slide.• Each of the four steps needs to be explained
plainly with a lot of words: Complete thoughts and complete sentences.
• The final statement is in plain English, suitable for the general public to understand.
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The Parking Ticket problem done as an essay question
1. State the hypotheses• We investigate the claim
that the average student receives more than nine parking tickets in a semester. Our hypotheses are:
• Null hypothesis, H0: μ ≤ 9• Alternative hypothesis:
Ha: μ > 9, more than nine tickets per semester.
2. Find the critical value• This is a t Test, right tailed.• The sample size is n = 27.• The degrees of freedom is
d.f. = n – 1 = 26.• The level of significance
chosen is α = 0.10• The critical value is
tα=0.10,d.f.=26 = 1.315
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The Parking Ticket problem done as an essay question
3. Compute the test statistic• (As shown on the earlier
slide – formula & details)
4. Conclusions• Since the test value 2.771 is
greater than the critical value 1.315, we reject the null hypothesis.
• “There is sufficient evidence to support the claim that the average student gets more than 9 parking tickets per semester.”
(Added content by D.R.S.)
2.771
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Draw a conclusion:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
A hometown department store has chosen its marketing strategies for many years under the assumption that the average shopper spends no more than $100 in their store. A newly hired store manager claims that the current average is higher, and wants to change their marketing scheme accordingly. A group of 24 shoppers is chosen at random and found to have spent on average $104.93 with a standard deviation of $9.07. Test the store manager’s claim at the 0.010 level of significance.
Solution:
First state the hypotheses:H0:Ha:
Next, set up the hypothesis test and determine the critical value: d.f. = 23, a = 0.010t0.010 =Reject if t ≥ t , or if t > 2.500.
m ≤ 100m > 100
2.500
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Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
Solution (continued):
Gather the data and calculate the necessary sample statistics:n = 24, = 100, = 104.93, s = 9.07,
Finally, draw a conclusion:Since t is greater than t a , we will reject the null hypothesis.
2.663
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Added content
• Repeating several of the slides with extra comments about TI-84
• Also an important reminder: using this method for small sample sizes requires that the population being studied is NORMALLY DISTRIBUTED. Not uniform, not skewed, but a bell curve distribution is assumed. (This book somewhat glosses over this point.
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Find the critical value:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
Find the critical t-score for a right-tailed test that has 14 degrees of freedom at the 0.025 level of significance.
Solution:d.f. = 14 and a = 0.025t0.025 = 2.145
The critical values for n < 30 are found from the t-distribution.
invT(area to left, d.f.) = t valuePlus or Minus Sign? Either by symmetry or by adjusting the area value for a right-tailed test.You still have to understand whether it’s left-tailed, right-tailed, or two-tailed. The calculator won’t do that for you !
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Draw a conclusion:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
27 college students are asked how many parking tickets they get a semester to test the claim that the average student gets more than 9 tickets a semester. The sample mean for the number of tickets was found to be 9.8 with a standard deviation of 1.5. Use a = 0.10.
Solution:
n = 27, = 9, = 9.8, s = 1.5, d.f. = 26, a = 0.10t0.10 = 1.315
Again, fix up the sign by knowing that it’s a right-tailed test, therefore positive critical value. The calculator will not do this thinking for you.
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Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
(continued from previous slide)
Solution:
n = 27, = 9, = 9.8, s = 1.5, d.f. = 26, a = 0.10
t0.10 =
Since t is greater than t a , we will reject the null hypothesis.
1.315
2.771
EXTRA ( ) around complicated numerators and denominators !!!
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Draw a conclusion:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
A hometown department store has chosen its marketing strategies for many years under the assumption that the average shopper spends no more than $100 in their store. A newly hired store manager claims that the current average is higher, and wants to change their marketing scheme accordingly. A group of 24 shoppers is chosen at random and found to have spent on average $104.93 with a standard deviation of $9.07. Test the store manager’s claim at the 0.010 level of significance.
Solution:
First state the hypotheses:H0:Ha:
Next, set up the hypothesis test and determine the critical value: d.f. = 23, a = 0.010t0.010 =Reject if t ≥ t , or if t > 2.500.
m ≤ 100m > 100
2.500
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Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
Solution (continued):
Gather the data and calculate the necessary sample statistics:n = 24, = 100, = 104.93, s = 9.07,
Finally, draw a conclusion:Since t is greater than t a , we will reject the null hypothesis.
2.663
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TI-84 T-Test
• The TI-84 has a built in Hypothesis Testing tool• STAT menu, TESTS submenu, 2:T-Test• You must understand how to do hypothesis
testing with charts and formulas, however. The calculator is not a substitute for that. Mere button smashing will lead you to failure.
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Example:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
27 college students are asked how many parking tickets they get a semester to test the claim that the average student gets more than 9 tickets a semester. The sample mean for the number of tickets was found to be 9.8 with a standard deviation of 1.5. Use a = 0.10.
Solution: Choose “Data” if the 27 data values were in TI-84 Lists,Stats if we have summary statistics already calculated
Null hypothesis’s mean
Sample’sMean, Standard deviation, and Size Direction of the
Alternative Hypothesis
Highlight “Calculate” and press ENTER
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Example, continued:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples). . . . Use a = 0.10.
Verify that it did theTest you wanted and that it has the correct Alternative Hypothesis. Verify that the sample data is correct.
The t= is the Test Statistic. It comes from the same formula as the one we’ve been using.
The p = is the p-value. It is the area to the right of that t value (in the case of this right-tailed test.) It is the probability of getting a t value as extreme as the t value we got.
When using the calculator’s T-Test, we use the “p-value method”. You don’t need a t critical value. Instead, you compare your p-value to the α (alpha) level of significance. If your p < α(alpha), thenthe decision is “Reject H0”.
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The other example, done with TI-84 T-Test and the p-value method:
Hypothesis Testing
10.2 Hypothesis Testing for Means
(Small Samples)
A hometown department store has chosen its marketing strategies for many years under the assumption that the average shopper spends no more than $100 in their store. A newly hired store manager claims that the current average is higher, and wants to change their marketing scheme accordingly. A group of 24 shoppers is chosen at random and found to have spent on average $104.93 with a standard deviation of $9.07. Test the store manager’s claim at the 0.010 level of significance.
H0:Ha:
m ≤ 100
m > 100
Compare your p-value p=.0069501788 to alpha: α=0.010and make the decision: Should we reject H0?