Question

The decrease in cholesterol level after eating a certain brand of oatmeal for breakfast for one month in people with cholesterol levels is Normally distributed. A random sample of 25 adults was selected and mean 8.5 and standard deviation 3. The brand advertises that eating its oatmeal for breakfast daily for one month will produce a mean decrease in cholesterol of more than 10 units for people with cholesterol levels. (α = 0.05)

a. Is there enough evidence to suggest that the mean decrease in cholesterol is actually less than advertised?

b. Given your conclusion, are you at risk of making a Type I error or a Type II error? Provide your answer in the context of the problem.

c. Report a 95% one-sided confidence interval for the mean decrease in cholesterol.

d. What is the power of the test?

Answer 1

a)

Hypothesis

The Null and Alternative Hypotheses are defined as,

This is a right-tailed test.

The significance level = 0.05

Test-statistic

The t statistic is obtained using the formula,

P-value

The p-value is obtained from t distribution table for t = -2.5 and degree of freedom = n - 1 = 25 - 1 = 24. (In excel use function =T.DIST.RT(-2.5,24))

Conclusion

Since the P-value is greater than the significance level = 0.05 at a 5% significance level, the null hypothesis is not rejected. Hence we can conclude that there is not sufficient evidence to conclude that the mean decrease in cholesterol is less than advertised.

b)

Answer: Type II error

Explanation: A Type II error is the probability of failing to reject the null hypothesis when the null hypothesis is actually FALSE. Since we are failed to reject the null hypothesis, there is the possibility of a type II error.

c)

The one-sided 95% confidence interval for the mean is obtained using the formula,

The t critical value is obtained from t distribution table for significance level = 0.05 and degree of freedom = n -1 = 125 - 1 = 24.

d)

R code:

power.t.test(n=25,delta=-1.5,sd=3,sig.level = 0.05,power = NULL,type = "one.sample", alternative = "one.sided", strict = FALSE)

R output

Explanation: