7. A coffee manufacturer is interested in whether the mean daily consumption of
ID: 3310224 • Letter: 7
Question
7. A coffee manufacturer is interested in whether the mean daily consumption of regular-coffee drinkers is less than that of decaffeinated-coffee drinkers. Assume the population standard deviation for those drinking regular coffee 1.20 cups per day and 1.36 cups per day for those drinking decaffeinated coffee. A random sample of 50 drinkers showed a mean of 4.35 cups per day. A sample of 40 decaffeinated-coffee drinkers showed a mean of 5.84 cups per day. Use the 0.01 significance level. Can they conclude that the mean daily consumption of regular- coffee drinkers is less than that of decaffeinated-coffee drinkers?Explanation / Answer
Solution:-
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: Regular> Decaffinated
Alternative hypothesis: Regular < Decaffinated
Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the mean difference between sample means is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.01. Using sample data, we will conduct a two-sample t-test of the null hypothesis.
Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).
SE = sqrt[(s12/n1) + (s22/n2)]
SE = 0.274
DF = 88
t = [ (x1 - x2) - d ] / SE
t = - 5.44
where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.
The observed difference in sample means produced a t statistic of - 5.43. We use the t Distribution Calculator to find P(t < - 5.43)
Therefore, the P-value in this analysis is less than 0.0001
Interpret results. Since the P-value (almost 0) is less than the significance level (0.01), we have to reject the null hypothesis.
From the above test we have sufficient evidence in the favor of tha claim that the mean daily consumption of the regular drinkers is less than that of decaffienated-coffee drinkers.
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