1. Tests of Hypotheses:
Small Samples
Chapter
Rejection
region

2. CHAPTER GOALS
TO DESCRIBE THE MAJOR CHARACTERISTICS OF STUDENT’S t-DISTRIBUTION.
TO UNDERSTAND THE DIFFERENCE BETWEEN THE t -DISTRIBUTION AND THE z -DISTRIBUTION.

3. CHAPTER GOALS TO TEST A HYPOTHESIS INVOLVING ONE POPULATION MEAN.
TO TEST A HYPOTHESIS INVOLVING THE DIFFERENCE BETWEEN TWO POPULATION MEANS.
TO CONDUCT A TEST OF HYPOTHESIS FOR THE DIFFERENCE BETWEEN A SET OF PAIRED OBSERVATIONS.

4. CHARACTERISTICS OF STUDENT’S t-DISTRIBUTION
The t-distribution has the following properties:
It is continuous, bell shaped and symmetrical about zero like the z-distribution.
There is a family of t-distributions with mean of zero but one for each sample size.
The t-distribution is more spread out and flatter at the center than the z-distribution, but approaches the z-distribution as the sample size gets larger.

5. z-distribution t-distribution The degrees of freedom for
the t-distribution
is df = n - 1.
z-distribution
t-distribution

6. A TEST FOR A POPULATION MEAN: SMALL SAMPLE, POPULATION STANDARD DEVIATION UNKNOWN
The test statistic for a one sample case is given by equation (9-1) below
(9-1)

7. EXAMPLE 1
The current rate for producing 5 amp fuses at Monarch Electric Company is 250 per hour. A new machine has been purchased and installed that, according to the supplier, will increase the production rate. A sample of 10 randomly selected hours from last month revealed the mean hourly production on the new machine was 256, with a sample standard deviation of 6 per hour. At the 0.05 significance level can Monarch conclude that the new machine is faster?

8. EXAMPLE 1 (continued)
Step 1: State the null and the alternative hypotheses.
H0: m £ H1: m > 250
Step 2: State the decision rule.
H0 is rejected if t > 1.833, df = 9 (Appendix F).
Step 3: Compute the value of the test statistic.
t = [ ]/[6/Ö10] = 3.16.
Step 4: What is the decision on H0?
H0 is rejected. The new machine is faster.

10. Display of the Rejection Region, Critical Value, and the computed Test Statistic
df = 9
Critical value
1.833
Region of
rejection
0.05 t

11. COMPARING TWO POPULATIONS MEANS
To conduct this test, three assumptions are required:
1. The populations must be normally or approximately normally distributed.
2. The populations must be independent.
3. The population variances must be equal.
Let subscript 1 and 2 be associated with population 1 and 2 respectively.

12. POOLED SAMPLE VARIANCE AND TEST STATISTIC
(9 -2)
(9 -3)

13. EXAMPLE 2
A recent Environmental Protection Agency (EPA) study compared the highway fuel economy of domestic and imported passengers cars. A sample of 15 domestic cars revealed a mean of 33.7 mpg with a standard deviation of 2.4 mpg. A sample of 12 imported cars revealed a mean of 35.7 mpg with a standard deviation of At the 0.05 significance level can the EPA conclude that the miles per gallon is higher on the imported cars? (Let subscript 1 be associated with the domestic cars).

14. EXAMPLE 2 (continued)
Step 1: State the null and the alternative hypotheses.
H0: m2 £ m H1: m2 > m1
Step 2: State the decision rule.
H0 is rejected if t > 1.708, df = 25.
Step 3: Compute the value of the test statistic.
t = 1.64 (Verify).
Step 4: What is the decision on H0?
H0 is not rejected. Insufficient sample evidence to claim a higher mpg on the imported cars.

16. Sampling Distribution for the Statistic t for a Two-Tailed Test, 0
Sampling Distribution for the Statistic t for a Two-Tailed Test, 0.05 Level of Significance
df = 25
Critical
value
-2.06
Critical
value
2.06
Do not
reject H0
Region of
rejection
Region of
rejection
0.95
0.025
0.025 t
-2.06
2.06

17. HYPOTHESIS TESTING INVOLVING PAIRED OBSERVATIONS
Use the following test when the samples are dependent.
For example, suppose you were collecting data on the price charged by two different body shops because you suspect that one is charging more than the other.
In this case, the same wrecked vehicle will be assessed by the two shops.
Because of this, the samples will be dependent.
Here we will take the difference of the two estimates and perform a test on the differences.

18. TEST STATISTIC (9 - 4) d-bar is the average of the differences
sd is the standard deviation of the differences
n is the number of pairs (differences)

19. EXAMPLE 3
An independent testing agency is comparing the daily rental cost for renting a compact car from Hertz and Avis. A random sample of eight cities is obtained and the following rental information obtained. At the 0.05 significance level can the testing agency conclude that there is a difference in the rental charged?
NOTE: These samples are dependent since the same type of car (compact) is being rented from the two companies in the same cities.

20. EXAMPLE 3 (continued)

21. EXAMPLE 3 (continued)
State the null and the alternative hypotheses:
H0:md = H1:md ¹ 0
State the decision rule.
H0 is rejected if t < or t >
Compute the value of the test statistic.
t = (1.00)/[3.162/Ö8] = 0.89, (verify).
What is the decision on the null hypothesis?
H0 is not rejected. There is no difference in the charge.

23. t df = 7 2.365 -2.365 Computed t = 0.89 Rejection Rejection region
0.89

24. Homework for Chapter 11,12,13
Chapter 11 : CD-ROM
Chapter 12 : CD-ROM
Chapter 13 : CD-ROM