Question

Crescent Oil has developed three new blends of gasoline – Blend X, Blend Y and Blend Z, and must decide which blend or blends to produce and distribute. A study of the miles per gallon ratings of the three blends is being conducted to determine if the mean ratings are the same for the three blends.

Five automobiles- 1, 2 3, 4 and 5, have been tested using each of the three gasoline blends and the miles per gallon ratings are shown on the accompanying Excel spreadsheet.

Based on the sample data we would like to determine whether the different blends of gasoline, produce significant differences in the average mpg. We would like to use the methods we have learnt so far in 361A to see if our result is statically significant. (Statistical significance refers to a result that is not likely to occur randomly but rather is likely to be attributable to a specific cause – in this case the different gasoline blends and different cars.)

Carry out the following tests and make preliminary findings:

1. For the sample data, calculate the means and standard deviations for the mpg for each of the three blends of gasoline – Blend X, Blend Y and Blend Z.
2. Draw three boxplots using Excel for mpg, one for each blend of gasoline.
3. Run three two sample t-tests between the different blends of gasoline i.e. compare the means of Blend X and Blend Y, then Blend X and Blend Z and finally Blend Y and Blend Z. Are they the same or are they different?
4. Armed with this information above determine whether or not your results show that the three different gasoline blends produce the same average mpg or not. Your answers should specifically site the information you re using to make your determination.
5. Re-run parts 1 - 4 above but this time your focus is on the mpg of each car for the 5 cars (car 1 through car 5) not for the three gasoline blends.

Your report should have the following sections, arranged sequentially:

1. Introduction and problem background

2. Data description and the business questions to be answered

3. Initial data exploration – descriptive statistics/graphs

4. Analyses

5. Interpretation of results, deficiencies in methods, final conclusions and recommendations for decision-making

Automobile	Blend X	Blend Y	Blend Z
1	31	30	30
2	30	29	29
3	29	29	28
4	33	31	29
5	26	25	26

Answer 1

The boxplots are:

For Brand X and Brand Y:

The hypothesis being tested is:

H0: µ1 = µ2

H1: µ1 ≠ µ2

Blend X	Blend Y
29.80	28.80	mean
2.59	2.28	std. dev.
5	5	n
	8	df
	1.000	difference (Blend X - Blend Y)
	5.950	pooled variance
	2.439	pooled std. dev.
	1.543	standard error of difference
	0	hypothesized difference
	0.648	t
	.5350	p-value (two-tailed)

The p-value is 0.5350.

Since the p-value (0.5350) is greater than the significance level (0.05), we cannot reject the null hypothesis.

Therefore, we can conclude that gasoline blends produce the same average mpg.

For Brand X and Brand Z:

The hypothesis being tested is:

H0: µ1 = µ2

H1: µ1 ≠ µ2

Blend X	Blend Z
29.80	28.40	mean
2.59	1.52	std. dev.
5	5	n
	8	df
	1.400	difference (Blend X - Blend Z)
	4.500	pooled variance
	2.121	pooled std. dev.
	1.342	standard error of difference
	0	hypothesized difference
	1.043	t
	.3272	p-value (two-tailed)

The p-value is 0.3272.

Since the p-value (0.3272) is greater than the significance level (0.05), we cannot reject the null hypothesis.

Therefore, we can conclude that gasoline blends produce the same average mpg.

For Brand Y and Brand Z:

The hypothesis being tested is:

H0: µ1 = µ2

H1: µ1 ≠ µ2

Blend Y	Blend Z
28.80	28.40	mean
2.28	1.52	std. dev.
5	5	n
	8	df
	0.400	difference (Blend Y - Blend Z)
	3.750	pooled variance
	1.936	pooled std. dev.
	1.225	standard error of difference
	0	hypothesized difference
	0.327	t
	.7524	p-value (two-tailed)

The p-value is 0.7524.

Since the p-value (0.7524) is greater than the significance level (0.05), we cannot reject the null hypothesis.

Therefore, we can conclude that gasoline blends produce the same average mpg.

The results show that the three different gasoline blends produce the same average mpg.

Anova: Two-Factor Without Replication
SUMMARY	Count	Sum	Average	Variance
1	3	91	30.33333	0.333333
2	3	88	29.33333	0.333333
3	3	86	28.66667	0.333333
4	3	93	31	4
5	3	77	25.66667	0.333333
Blend X	5	149	29.8	6.7
Blend Y	5	144	28.8	5.2
Blend Z	5	142	28.4	2.3
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Rows	51.33333	4	12.83333	18.78049	0.000396	3.837853
Columns	5.2	2	2.6	3.804878	0.068988	4.45897
Error	5.466667	8	0.683333
Total	62	14