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The paper "Does the Color of the Mug Influence the Taste of the Coffee?" (Flavour [2014]: 1-7) describes an experiment in which subjects were assigned at random to one of two treatment groups. The 12 people in one group were scrved coffee in a white mug and were asked to rate the quality of the coffee on a scale from 0 to 100 . The 12 people in the second group were served the same coffee in a clear glass mug, and they also rated the coffee. The mean quality rating for the 12 people in the white mug group was 50.35 and the standard deviation was \(20.17 .\) The mean quality rating for the 12 people in the clear glass mug group was 61.48 and the standard deviation was \(16.69 .\) For purposes of this exercise, you may assume that the distribution of quality ratings for each of the two treatments is approximately normal. a. Use the given information to construct and interpret a \(95 \%\) confidence interval for the difference in mean quality rating for this coffee when served in a white mug and when served in a glass mug. b. Based on the interval from Part (a), are you convinced that the color of the mug makes a difference in terms of mean quality rating? Explain.

Step by step solution

01

Identify the formulas

In this step, we will identify the formulas needed to calculate the confidence interval. 1. Standard Error (SE) of the difference = \(\sqrt{\frac{s_{1}^2}{n_1} + \frac{s_{2}^2}{n_2}}\) 2. Margin of Error (ME) = critical value * Standard Error 3. Confidence Interval = (Difference in Means - Margin of Error, Difference in Means + Margin of Error) Here, \(s_{1}\) represents the standard deviation of white mug group, \(s_{2}\) represents the standard deviation of glass mug group, \(n_1\) represents the sample size of white mug group (12), and \(n_2\) represents the sample size of the glass mug group (12)

02

Calculate the Standard Error

Use the standard error formula to calculate the standard error of the difference in mean quality ratings. SE = \(\sqrt{\frac{20.17^2}{12} + \frac{16.69^2}{12}}=5.717\)

03

Calculate the Margin of Error

For a 95% confidence interval, the critical value is approximately 1.96. Calculate the margin of error using the critical value and the standard error. ME = 1.96 * 5.717 = 11.206

04

Calculate the Confidence Interval

Now, calculate the confidence interval using the difference in means and the margin of error. Confidence Interval = (50.35 - 61.48 - 11.206, 50.35 - 61.48 + 11.206) = (-22.336, 0.924) #b. Concluding if the color of the mug makes a difference#

05

Interpret the Confidence Interval

The confidence interval we constructed is (-22.336, 0.924). Since it includes zero, we cannot conclude with 95% confidence that the mug's color affects the mean quality rating of the coffee. This means we do not have strong evidence to suggest that the color of the mug makes a significant difference in terms of mean quality rating.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Experimental Design

Experiments like the one described in the exercise aim to uncover relationships between different variables by observing outcomes under controlled conditions. In this specific experiment, researchers wanted to find out if the color of a mug could influence the perception of coffee taste.

Key features of experimental design include:

• Random Assignment: Subjects were randomly assigned to two groups to ensure any differences in results are due to the color of the mug and not other factors.
• Controlled Variables: All other variables such as the type of coffee and environment were kept constant, to focus solely on the impact of the mug's color.
• Sample Size: Each group consisted of 12 people, making it 24 in total, which is a small sample size but allows preliminary insights.

These elements help ensure the reliability and validity of the experiment's findings.

Standard Error

Standard Error (SE) measures how much difference exists between a sample statistic and the actual population parameter. It's pivotal in understanding the precision of sample data when estimating population metrics. SE is crucial in this experiment because it shows the accuracy of the mean quality ratings.To calculate SE for the difference between two groups, the formula is:\[SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\]- Here, \(s_1\) and \(s_2\) are the standard deviations of each group, and \(n_1\) and \(n_2\) are the sample sizes.A smaller SE indicates that the sample mean is a more reliable estimate of the true population mean. In this case, the calculated SE is \(5.717\). This means there's a moderate level of variability in how accurately the sample represents the overall population.

Mean Quality Rating

The mean quality rating provides a numerical average that reflects the general perception of the coffee's quality among participants. In this study, the mean rating differs between the two groups, with the white mug group having a mean of 50.35, and the glass mug group a mean of 61.48.
Understanding this concept involves:

• Mean Calculation: The mean is calculated by summing all individual ratings in a group and dividing by the number of participants (12 in each group).
• Implication of Means: At first glance, it might seem that coffee in a glass mug is perceived as better, due to a higher mean rating.

However, mean alone doesn’t indicate if the difference is statistically significant. That's where further statistical methods, like confidence intervals, come into play.

Statistical Significance

Statistical significance indicates whether a result is likely due to a specific factor rather than random chance. In this study, the researchers use a confidence interval to support conclusions about statistical significance between coffee mug colors and taste perceptions. Here's how this works:

• Confidence Interval: The interval calculated is (-22.336, 0.924). Since this includes zero, it suggests that the observed difference in means might be due to random variability rather than an actual effect of mug color.
• Significance Level: Typically set at 0.05 for a 95% confidence interval, indicating that a result would be considered significant if the confidence interval did not include zero.

Ultimately, the findings imply that, based on the study's data and design, there isn't enough evidence at the 95% confidence level to confirm that mug color significantly influences coffee taste perception.