Question

A cereal manufacturer has advertised that its product, Fiber Oat Flakes, has a lower fat content than its competitor, Bran Flakes Plus. Because of complaints from the manufacturers of Bran Flakes Plus, the FDA has decided to test the claim that Oat Flakes has a lower average fat content than Bran Flakes Plus. Several boxes of each cereal are selected and the fat content per serving is measured. The results of the study are as follows.

	N	Y-bar	S
Fiber Oat Flakes	16	5	1
Bran Flakes Plus	15	6	1

1. Identify the response variable and explanatory variable. Also classify each as categorical or quantitative.
2. Is this an observational study or an experiment? Explain.
3. State the parameter of interest in WORDS.
4. Report the null and alternative hypotheses in symbols.
5. Find the estimated difference of the mean fat content between the two groups.
6. Find the standard error of the estimated differences.
7. At alpha=0.05, is mean fat content of Oat Flakes brand is significantly lower than the mean fat content of Bran Flakes Plus? Compute the –t-statistic.
8. Assuming the normality assumption is satisfied, knowing that degrees of freedom from JMP is 28. Approximate the pvalue the test. And state your decision.
9. State your conclusion.
10. Construct and interpret a 95% Confidence interval of the difference of mean scores between the groups (use by df from JMP df= 28)
11. Based on your confidence interval what does it say about the null hypothesis that the difference of the means is 0?
12. Assume you are repeating this experiment with only 5 boxes in each group (but the means and std. dev. are as before). Do you think the decision of the test will change or not? If yes, how? If not, how? Explain your answer.
13. Do the sample data provide evidence that the mean fat content of Bran Flakes is very different than the mean fat content of Oat Flakes? Explain.

Answer 1

The response variable is average fat content . This is a quantitative variable.

The explanatory variable is cereal manufacturer. This is a categorical variable with two values - Fiber Oat Flakes and Bran Flakes Plus.

The the parameter of interest is average fat content in a cereal box.

Null hypothesis H0: Average fat content in Oat Flakes is equal to the average fat content in a Bran Flakes Plus box.

Alternative hypothesis H0: Average fat content in Oat Flakes is less than the average fat content in a Bran Flakes Plus box.

Estimated difference of the mean fat content between the two groups = 5 - 6 = -1

Standard error of the estimated differences. = = 0.3594

t-statistic = Estimated difference of the mean / Standard error of the estimated differences

= -1/0.3594

= -2.78

P-value = P[t < -2.78, df = 28] = 0.0048

Since p-value is less than 0.05 significance level, we reject null hypothesis H0 and conclude that there is significant evidence that average fat content in Oat Flakes is less than the average fat content in a Bran Flakes Plus box.

Critical value of t at 95% Confidence interval and df = 28 is 2.05

95% Confidence interval of the difference of mean scores between the group (Fiber Oat Flakes - Bran Flakes Plus)

= (-1 - 2.05 * 0.3594 , -1 + 2.05 * 0.3594)

= (-1.73677 , -0.26323)

Since the 95% Confidence interval does not contain the value 0,  we reject the null hypothesis that the difference of the means is 0.

Standard error of the estimated differences. = = 0.6325

t-statistic = Estimated difference of the mean / Standard error of the estimated differences

= -1/0.6325

= -1.58

P-value = P[t < -1.58, df = 28] = 0.0627

Since p-value is greater than 0.05 significance level, we fail to reject null hypothesis H0 and conclude that there is no significant evidence that average fat content in Oat Flakes is less than the average fat content in a Bran Flakes Plus box.

Yes, the decision of the test will change.

From initial sample sizes, p-value for two-tail test is 2 * 0.0048 = 0.0096

Since p-value is less than 0.05 significance level, we conclude that there is significant evidence that the mean fat content of Bran Flakes is very different than the mean fat content of Oat Flakes