Math  /  Data & Statistics

QuestionUse a tt-distribution to find a confidence interval for the difference in means μd=μ1μ2\mu_{d}=\mu_{1}-\mu_{2} using the relevant sample results from paired data. Assume the results come from random samples from populations that are approximately normally distributed, and that differences are computed using d=x1x2d=x_{1}-x_{2}.
A 95\% confidence interval for μd\mu_{d} using the paired data in the following table: \begin{tabular}{l|ll} \hline Case & \begin{tabular}{l} Situation \\ 1 \end{tabular} & \begin{tabular}{l} Situation \\ 2 \end{tabular} \\ \hline 1 & 78 & 86 \\ 2 & 80 & 85 \\ 3 & 95 & 90 \\ 4 & 62 & 78 \\ 5 & 71 & 78 \\ 6 & 72 & 62 \\ 7 & 84 & 88 \\ 8 & 91 & 92 \\ \hline \end{tabular}
Give the best estimate for μd\mu_{d}, the margin of error, and the confidence interval. Enter the exact answer for the best estimate, and round your answers for the margin of error and the confidence interval to two decimal places. best estimate == \square 3.5-3.5 margin of error = \square \square 6.71-6.71
The 95%95 \% confidence interval is to i 3.21 \square

Studdy Solution

STEP 1

1. The data comes from random samples from populations that are approximately normally distributed.
2. We are using paired data, where differences are computed using d = x₁ - x₂.
3. We will use a t-distribution for the confidence interval.
4. The confidence level is 95%.

STEP 2

1. Calculate the differences between paired data points.
2. Compute the mean of the differences.
3. Calculate the standard deviation of the differences.
4. Determine the degrees of freedom and t-value.
5. Calculate the margin of error.
6. Compute the confidence interval.

STEP 3

Calculate the differences (d = x₁ - x₂) for each pair:
\begin{align*} d_1 &= 78 - 86 = -8 \\ d_2 &= 80 - 85 = -5 \\ d_3 &= 95 - 90 = 5 \\ d_4 &= 62 - 78 = -16 \\ d_5 &= 71 - 78 = -7 \\ d_6 &= 72 - 62 = 10 \\ d_7 &= 84 - 88 = -4 \\ d_8 &= 91 - 92 = -1 \end{align*}

STEP 4

Compute the mean of the differences:
dˉ=8+(5)+5+(16)+(7)+10+(4)+(1)8=3.25 \bar{d} = \frac{-8 + (-5) + 5 + (-16) + (-7) + 10 + (-4) + (-1)}{8} = -3.25
This is the best estimate for μd.

STEP 5

Calculate the standard deviation of the differences:
sd=(didˉ)2n1 s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n - 1}}
First, we need to calculate (d_i - \bar{d})^2 for each difference:
\begin{align*} (-8 - (-3.25))^2 &= (-4.75)^2 = 22.5625 \\ (-5 - (-3.25))^2 &= (-1.75)^2 = 3.0625 \\ (5 - (-3.25))^2 &= (8.25)^2 = 68.0625 \\ (-16 - (-3.25))^2 &= (-12.75)^2 = 162.5625 \\ (-7 - (-3.25))^2 &= (-3.75)^2 = 14.0625 \\ (10 - (-3.25))^2 &= (13.25)^2 = 175.5625 \\ (-4 - (-3.25))^2 &= (-0.75)^2 = 0.5625 \\ (-1 - (-3.25))^2 &= (2.25)^2 = 5.0625 \end{align*}
Now, sum these values and divide by (n-1):
sd=22.5625+3.0625+68.0625+162.5625+14.0625+175.5625+0.5625+5.06257=451.578.0320 s_d = \sqrt{\frac{22.5625 + 3.0625 + 68.0625 + 162.5625 + 14.0625 + 175.5625 + 0.5625 + 5.0625}{7}} = \sqrt{\frac{451.5}{7}} \approx 8.0320

STEP 6

Determine the degrees of freedom and t-value: - Degrees of freedom: df = n - 1 = 8 - 1 = 7 - For a 95% confidence interval, we need t₀.₀₂₅ with 7 degrees of freedom - From a t-distribution table or calculator, we find: t₀.₀₂₅,₇ ≈ 2.365

STEP 7

Calculate the margin of error:
Margin of Error=t0.025,7sdn=2.3658.032086.7138 \text{Margin of Error} = t_{0.025,7} \cdot \frac{s_d}{\sqrt{n}} = 2.365 \cdot \frac{8.0320}{\sqrt{8}} \approx 6.7138
Rounded to two decimal places: 6.71

STEP 8

Compute the confidence interval:
Lower bound = \bar{d} - \text{Margin of Error} = -3.25 - 6.71 = -9.96 Upper bound = \bar{d} + \text{Margin of Error} = -3.25 + 6.71 = 3.46
Rounded to two decimal places: 95% Confidence Interval: (-9.96, 3.46)
The results are: Best estimate for μd = -3.25 Margin of error = 6.71 95% Confidence Interval: (-9.96, 3.46)

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