A study investigated whether price affects people's judgment. Twenty people each tasted six cabernet
sauvignon wines and rated how they liked them on a scale of 1 to 6. Prior to tasting each wine, participants
were told the price of wine. Of the six wines tasted, two were actually the same wine, but for one tasting
the participant was told that the wine cost $10 per bottle and for the other tasting the participant was told
that the wine cost $90 per bottle. The participants were randomly assigned to either taste the wine claimed
to be $90 first and the wine claimed to be $10 second or the $10 wine first and the $90 wine second.
Differences were calculated by subtracting the rating for the tasting in which the participant thought the
wine cost $10 from the rating for the tasting in which the participant thought the wine cost $90. The
differences that follow are consistent with summary quantities given in the paper.
Difference in Rating ($90 − $10)
2 4 3 2 1 0 0 3 0 2 2 3 3 1 4 1 3 −1 1 −1
Use the data given above to construct a 90% confidence interval for the difference in mean rating assigned
to the wine when the cost is described as $90 and when the cost is described as $10. (Use $90 − $10.
Enter your answer using interval notation. Round your numerical values to four decimal places.)
Guide On Rating System
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First, we need to calculate the sample mean and standard deviation of the differences.
Sample mean:
$\bar{x}=\frac{2+4+3+2+1+0+0+3+0+2+2+3+3+1+4+1+3-1+1-1}{20}=\frac{32}{20}=1.6$
Sample standard deviation:
$s=\sqrt{\frac{\sum(x_i-\bar{x})^2}{n-1}}=\sqrt{\frac{(2-1.6)^2+(4-1.6)^2+(3-1.6)^2+(2-1.6)^2+(1-1.6)^2+(0-1.6)^2+(0-1.6)^2+(3-1.6)^2+(0-1.6)^2+(2-1.6)^2+(2-1.6)^2+(3-1.6)^2+(3-1.6)^2+(1-1.6)^2+(4-1.6)^2+(1-1.6)^2+(3-1.6)^2+(-1-1.6)^2+(1-1.6)^2+(-1-1.6)^2}{19}}=\sqrt{\frac{10.4}{19}}\approx0.7456$
Next, we need to calculate the t-value for a 90% confidence interval with 19 degrees of freedom. By looking up the t-distribution table or using statistical software, we find that the t-value is approximately 1.729.
Finally, we can calculate the confidence interval using the formula:
Confidence interval = ($\bar{x} - t*\frac{s}{\sqrt{n}}$, $\bar{x} + t*\frac{s}{\sqrt{n}}$)
Plugging in the values, we get:
Confidence interval = (1.6 - 1.729 * $\frac{0.7456}{\sqrt{20}}$, 1.6 + 1.729 * $\frac{0.7456}{\sqrt{20}}$)
Simplifying, we get:
Confidence interval = (1.6 - 0.3178, 1.6 + 0.3178)
Confidence interval $\approx$ (1.2822, 1.9178)
Therefore, the 90% confidence interval for the difference in mean rating assigned to the wine when the cost is described as $90 and when the cost is described as $10 is approximately (1.2822, 1.9178).
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