Question

There has been an alcohol war between Bob and Len for some time now. The following results were observed. Does the data indicate that the average purchases at Len's exceed, on average, those at Bob's by more than 8.50 dollars? The data is given in the file below. Use α=0.072α=0.072 for all calculations.

And is summarized here

	Bobs Booze	Lens Liquor Emporium
n	46	39
average cost	50.02	59.56
variance of cost	10.15	3.6

(a) Form the correct hypothesis
A. H0:μLen=μBobHA:μLen<μBobH0:μLen=μBobHA:μLen<μBob
B. H0:μLen−μBob=8.5HA:μLen−μBob≠8.5H0:μLen−μBob=8.5HA:μLen−μBob≠8.5
C. H0:μLen−μBob=8.5HA:μLen−μBob>8.5H0:μLen−μBob=8.5HA:μLen−μBob>8.5
D. H0:μLen−μBob=8.5HA:μLen−μBob<8.5H0:μLen−μBob=8.5HA:μLen−μBob<8.5
E. H0:μLen=μBobHA:μLen>μBobH0:μLen=μBobHA:μLen>μBob
F. H0:μLen=μBobHA:μLen≠μBobH0:μLen=μBobHA:μLen≠μBob


(b) (i) Using technology available to you, test to see if the variances are equal or not?
A. They appear to be equal.
B. There appears to be more variation in Lens sales than in the Bobs.
C. There appears to be more variation in Bobs sales than in the Lens.


(b) (ii)Report the p-value of the test you ran in (b)(i) use at least three decimals in your answer.
P-value =

(c) Test the statistical hypotheses in (a) by carrying out the appropriate statistical test. Find the value of the test statistic for this test, use at least two decimals in your answer.
Value of the test statistic =

(d) Determine the PP-value for the difference of means test, and report it to at least three decimal places.
P=

(e) Determine the appropriate degrees of freedom for this test. As an integer.
DF=

(f) Based on the above calculations, we should  ? reject not reject  the null hypothesis.

Answer 1

Here we want to test whether the average purchases at Len's exceed, on average, those at Bob's by more than 8.50 dollars?

Using the above statement the correct null hypothesis and the alternative hypothesis are as follows:

H0: μLen − μBob = 8.5

HA: μLen − μBob > 8.5

So correct choice is option C

(b) (i) Using technology available to you, test to see if the variances are equal or not?

Let's use minitab:

Step 1) Click on Stat >>>Basic Statistics >>>2-Variances ...

Fill the necessary information and then click on Option again fill the necessary information.

Look the following image:

Then click on OK again Click on OK , so we get the following output:

(b) (ii)Report the p-value of the test you ran in (b)(i) use at least three decimals in your answer.

b ii) p-value = 0.001

Since p-value < 0.05 so reject the equality assumption of two populations.

So that we use unpooled two sample t test .

(c) Test the statistical hypotheses in (a) by carrying out the appropriate statistical test. Find the value of the test statistic for this test, use at least two decimals in your answer.

Now lets test the equality of two means using unpooled t test:

Level of significance = \alpha = 0.072 = 7.2%

therefore level of confidence = 100 - 7.2 = 92.8%

Standard deviation for Lens = SQRT(3.6) = 1.8973

Standard deviation for Bobs = SQRT(10.15) = 3.1859

Let's used minitab:

Steps 1) Click on Stat>>>Basic Statistics>>>2-Sample t...

Steps 1) Click on summarized data and then fill the required information in the boxes : look the following picture.

step 3) Click on Option, Look the following image :

then click on OK again click on OK

So we get the following output

Value of the test statistic =1.86

(d) Determine the P-value for the difference of means test, and report it to at least three decimal places.

P-value = 0.033

(e) Determine the appropriate degrees of freedom for this test. As an integer.

DF=74

(f) Based on the above calculations, we should  ? reject not reject  the null hypothesis.

Decision rule: 1) If p-value <= level of significance (alpha) then we reject null hypothesis

2) If p-value > level of significance (alpha) then we fail to reject null hypothesis.

Here p value = 0.033 < 0.072 so we used first rule.

That is we reject null hypothesis