Chapter 9Section 2

Testing the Difference Between Two Means: t Test

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9.2 Testing the Difference Between Two Means: Using the t Test

Formula for the t test for comparing two means from independent populations with unequal variances

where the degrees of freedom are equal to the smaller of n1 – 1 or n2 – 1.

2

1 2 1 2

2 21 2

1 2

X Xt

s sn n

Farm SizesThe average size of a farm in Indiana County, Pennsylvania, is 191 acres. The average size of a farm in Greene County, Pennsylvania, is 199 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 counties is different? Assume the populations are normally distributed.

Step 1: State the hypotheses and identify the claim.

H0: μ1 = μ2 and H1: μ1 μ2 (claim)

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≠

Farm SizesStep 2: Find the critical values.

Since the test is two-tailed, a = 0.05, and the variances are unequal, the degrees of freedom are the smaller of n1 – 1 or n2 – 1. In this case, the degrees of freedom are 8 – 1 = 7. Hence, from green page, the critical values are -2.365 and 2.365.

Step 3: Find the test value.

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1 2 1 2

2 21 2

1 2

X Xt

s sn n

2 2

191 199 0

38 128 10

0.57

Step 4: Make the decision.

Do not reject the null hypothesis.

Step 5: Summarize the results.

There is not enough evidence to support the claim that the average size of the farms is different.

Farm Sizes

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Confidence Intervals for the Difference Between Two Means

Formula for the t confidence interval for the difference between two means from independent populations with unequal variances

d.f. smaller value of n1 – 1 or n2 – 1.

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2 21 2

1 2 2 1 21 2

2 21 2

1 2 21 2

s sX X t

n n

s sX X t

n n

Example 9-5: Confidence IntervalsFind the 95% confidence interval for the difference between the means for the data in Farm Sizes.

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2 21 2

1 2 2 1 21 2

2 21 2

1 2 21 2

s sX X t

n n

s sX X t

n n

2 2

1 2

2 2

38 12191 199 2.365

8 1038 12

191 199 2.3658 10

1 241.0 25.0

Corn Yield

A farmer has agreed to plant two different types of test corn in his fields. Variety A was planted over 14 acres. Each acre was then sampled for yield. The average yield was 162 bushels to the acre with a sample standard deviation of 8.5 bushels. Variety B was planted over 10 acres. Again each acre was then sampled for yield and it 148 bushels to the acre with a sample standard deviation of 6.3 bushels. Can it be concluded at α = 0.05 that Sample A yields better than sample B?

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