Math Problem Statement

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Construct a 90% confidence interval for the population mean difference in tread wear between the two brands, based on the given data. Assume normal distribution of differences.

Solution

To construct the 90% confidence interval for the population mean difference (μd\mu_d), follow these steps:

Step 1: Calculate the Mean of the Differences

  1. List the differences: −0.3,0.34,−0.9,0.03,−1.03,0.12,−0.12,−0.15,−0.28,−0.02,−0.25,−0.74,−0.32-0.3, 0.34, -0.9, 0.03, -1.03, 0.12, -0.12, -0.15, -0.28, -0.02, -0.25, -0.74, -0.32.
  2. Calculate the mean (dˉ\bar{d}):

dˉ=(−0.3+0.34−0.9+0.03−1.03+0.12−0.12−0.15−0.28−0.02−0.25−0.74−0.32)13=−0.2723\bar{d} = \frac{(-0.3 + 0.34 - 0.9 + 0.03 - 1.03 + 0.12 - 0.12 - 0.15 - 0.28 - 0.02 - 0.25 - 0.74 - 0.32)}{13} = -0.2723

Step 2: Calculate the Standard Deviation of the Differences

  1. Calculate each difference from the mean, square them, and sum:
(-0.3 + 0.2723)^2 + \ldots + (-0.32 + 0.2723)^2 = 1.5799 \end{align*}$$ 2. Divide by $$n-1$$ to get the variance, then square root for standard deviation: $$s_d = \sqrt{\frac{1.5799}{12}} = 0.3649$$ ### Step 3: Find the Critical t-Value - Degrees of freedom = $$n-1 = 12$$ - Use a t-table or calculator for a 90% confidence interval: $$t_{0.05, 12} \approx 1.782$$ ### Step 4: Calculate the Margin of Error $$\text{Margin of Error} = t \times \frac{s_d}{\sqrt{n}} = 1.782 \times \frac{0.3649}{\sqrt{13}} = 0.1801$$ ### Step 5: Construct the Confidence Interval $$\text{Lower limit} = \bar{d} - \text{Margin of Error} = -0.2723 - 0.1801 = -0.4524$$ $$\text{Upper limit} = \bar{d} + \text{Margin of Error} = -0.2723 + 0.1801 = -0.0922$$ ### Conclusion The 90% confidence interval for the mean difference in tread wear is approximately: - Lower limit: $$-0.452$$ - Upper limit: $$-0.092$$

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Math Problem Analysis

Mathematical Concepts

Confidence Intervals
Descriptive Statistics
Hypothesis Testing

Formulas

Mean of differences (x̄_d) = (Σd_i) / n
Standard deviation (s_d) = √(Σ(d_i - x̄_d)^2 / (n - 1))
Margin of Error = t * (s_d / √n)

Theorems

Central Limit Theorem
t-Distribution for Small Samples

Suitable Grade Level

Grades 11-12 or College-Level Statistics

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