A matched pairs experiment compares the taste of instant coffee with fresh-brewed coffee. Each subject tastes two unmarked cups of coffee, one of each type, in random order and states which he or she prefers. Of the 50 subjects who participate in the study, 19 prefer the instant coffee and the other 31 prefer fresh-brewed. Let p be the proportion of the population that prefers fresh-brewed coffee. a. Test the claim that a majority of people prefer the taste of fresh-brewed coffee. Report the z test-statistic and its p-value. Is your test result significant at the α = 10% level? What is your practical conclusion? b. compute the 90% confidence interval. Use the data set above.
A matched pairs experiment compares the taste of instant coffee with fresh-brewed coffee. Each subject tastes two unmarked cups of coffee, one of each type, in random order and states which he or she prefers. Of the 50 subjects who participate in the study, 19 prefer the instant coffee and the other 31 prefer fresh-brewed.
Let p be the proportion of the population that prefers fresh-brewed coffee.
a. Test the claim that a majority of people prefer the taste of fresh-brewed coffee. Report the z test-statistic and its p-value. Is your test result significant at the α = 10% level? What is your practical conclusion?
b. compute the 90% confidence interval. Use the data set above.
Solution:
Population: • {Xi = 1} → Customer i prefers fresh-brewed coffee to instant coffee
• p = the proportion of the population that prefers fresh-brewed coffee.
Sample: size: n = 50 Data: Count: x = x1 + x2 +…+ x50 = 31
Sample Proportion Estimate: ?̅ = ? ? = ?? ?? = 0.62 (statistic)
X = Count how many customers prefers fresh-brewed coffee to instant coffee Estimator: Sample Proportion ?̅ = ? ?
➢ First verify that the
Since, n?̅ = ?? × ?. ?? = ?? ≥ 5 and n(1 –?̅) = 19 ≥ 5,
Then, ?̅~ Normal { p = ??? ; ???̅ = ____________ }
From Data: ➢ Sample Proportion estimate (statistic): ?̅ = ? ? = _____________
➢ Standard Error: ???̅ = √ ?̅ (?−?̅ ) ? = ____________________________
➢ Confidence level: C = 90%, → (α/2) = 0.05
➢ Critical Value: z* = _____________ (Use Z-Table or the last row of t-Table) ➢ Margin of error: m = z* × ???̅ = ______________________________________
➢ Confidence Interval: ?̅ ± m = _______________ = [ _____________ , ____________] ➢ RStudio OUTPUT: 90% confidence interval:
Lower confidence limit = ___________ Upper confidence limit = _______________
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