Aim: How do we test a comparison group?

Exam Tomorrow

test hypotheses that compare means of two groups

z test

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

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

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?

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

test the difference between two variances

F test

Formula for the F Test

• The larger of the two variances is placed in the numerator regardless of the subscripts

2122

sF

s

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

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

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?

Solution

test the difference between two means for small independent

samplest test

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

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

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

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

• 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