The document discusses using a t-test to test the difference between means of two independent samples. A t-test can be used when samples are from two normally distributed populations and are independent. The formula for a t-test calculates the test value as the difference between the observed and expected values divided by the standard error. An example compares the average farm sizes in two counties and performs a t-test to determine if the difference in means is statistically significant at the 0.05 level. The t-test results show there is not enough evidence to conclude the average farm sizes are different between the counties.

1. TESTING THE DIFFERENCE
BETWEEN TWO MEANS OF
INDEPENDENT SAMPLES:
USING THE T TEST

2. WHEN TO USE T TEST?
t test – used to test the
difference between means
when,
 the two samples are
independent
 When the samples are taken
from two normally or
approximately normally
distributed populations.

3.  Samples are independent
samples when they are not
related.
 Assume that the variances are
not equal.

4. FORMULA:

5. Test value= (observed value) –(expected
value)
standard error

6. Assumption for the t test for two
Independent Means when 𝝈1 and 𝝈2 are
unknown
1. The samples are random samples.
2. The sample data are independent of
one another.
3. When the sample sizes are less than
30, the populations must be normally
or approximately normally distributed.

10. EXAMPLE:
The 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 deviation 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.

11. STEP 1
State the hypotheses and identify the claim for the means
H0: 𝜇1= 𝜇2
H1: 𝜇1 ≠ 𝜇2 (claim)

12. STEP 2
Find the critical values. Since the test is
two-tailed, since 𝛼 = 0.05, and since the
variances are unequal,
Then,
 the degrees of freedom are the smaller
of n1 -1 or n2 -1

13. In this case, the degrees of freedom
are 8-1=7.
Hence, from the t table, the critical
values are +2.365 and – 2.365.

14. STEP 3

15. STEP 4
Make the decision. Do not reject the
null hypothesis, since -0.57 > -2.365.

16. STEP 5
 Summarize the results. There is not enough evidence to support the
claim that the average size of the farms is different.