aim: how do we test a comparison group? exam tomorrow
TRANSCRIPT
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Aim: How do we test a comparison group?
Exam Tomorrow
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test hypotheses that compare means of two groups
z test
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Formula for the z Test for Comparing Two Means from Independent
Populations
1 2 1 2
2 21 2
1 2
X Xz
n n
If you are comparing means, your will always equal 0 unless told otherwise.
1 2
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Procedure
1. State the hypotheses and identify the claim
2. Find the critical value(s)
3. Compute the test value
4. Make the decision
5. Summarize the results
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Example
• A survey found that the average hotel room rate in New Orleans is $88.42 and the average room rate in Phoenix is $80.61. Assume the data were obtained from two samples of 50 hotels each and the standard deviations were $5.62 and $4.83, respectively. At α = 0.05, can it be concluded that there is significant difference in the rates?
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Solution
1. State Hypothesis and identify claim
2. Find critical value
3. Calculate test value
4. Make a decision
5. Summarize the results
Test value > critical value (reject null)
There is enough evidence to support the claim that the means are not equal to each other
0 1 2
1 1 2
:
: ( )
H
H claim
.05
1.96
1 2 1 2
2 2 2 21 2
1 2
88.42 80.61 07.45
5.62 4.8350 50
X Xz
n n
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test the difference between two variances
F test
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Formula for the F Test
• The larger of the two variances is placed in the numerator regardless of the subscripts
2122
sF
s
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Characteristics of the F Distribution
1. The values of F cannot be negative, because variances are always positive or zero
2. The distribution is positively skewed
3. The mean value of F is approximately equal to 1
4. The F distribution is a family of curves based on the degrees of freedom of the variance of the numerator and the degrees of freedom of the variance of the denominator
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Steps
1. State the hypothesis and identify claim
2. Find the critical value
3. Compare the test values
4. Make the decision
5. Summarize the results
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Example
• The average size of a farm in Indiana County, Pennsylvania, is 191 acres. Assume the data were obtained from two samples with standard deviations of 38 and 12 acres, respectively, and sample sizes of 8 and 10, respectively. Using α = 0.05, is there enough evidence to support the claim that the variances are equal?
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Solution
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test the difference between two means for small independent
samplest test
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Use the t test when…
• Option 1– Used when the
variances of the populations are not equal
• Option 2– Use when the
variances are equal
If information about the variance is not given to you, you must use the F test and test the variance to see if they are equal or unequal
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Formula for the t test – for testing the difference between two means of a
small independent sample
• Variances are assumed to be unequal
1 2 1 2
2 21 2
1 2
X Xt
s sn n
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Formula for the t test – for testing the difference between two means of a
small independent sample
• Variances are assumed to be equal:
1 2 1 2
2 21 1 2 2
1 2 1 2
1 1 1 12
X Xt
n s n s
n n n n
Where the degree of freedom are equal to 1 2 2n n
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Procedure1. Need to determine if the variances are equal if they did
not give you any information about variances in the problem
1. State the hypothesis2. Find the critical value (use f table)3. Compute test value with f test4. Make a decision5. Summarize results
2. Decide which t test to use based on steps 1-51. State the hypothesis2. Find the critical values3. Compute test value with t test4. Make a decision5. Summarize results
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• The average size of a farm in Indiana County, Pennsylvania, is 191 acres. Assume the data were obtained from two samples with standard deviations of 38 and 12 acres, respectively, and sample sizes of 8 and 10, respectively. Can it be concluded at α = 0.05 that the average size of the farms in the two countries is different? Assume the populations are normally distributed.
Example
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