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Wize University Statistics Textbook > Match Pairs (Dependent Means)

Confidence Intervals for Matched Pairs

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Matched Pairs Confidence Interval for Differences in Dependent Means

We don't know what the actual value of Dˉ\bar D (or µdµ_d) is, but we can provide a reasonable range by constructing a confidence interval for the population average of the differences by using dˉ\bar d (or xˉdx̄_d) as a point estimate .
  • Confidence level C=1−αC=1-α
d‾±t∗(sdn)\boxed{\overline{d} \pm t^* \bigg(\dfrac{s_d}{\sqrt{n}}\bigg)}
where,
  • t∗t^* is based on df=n−1df = n-1
  • nn = number of pairs


Wize Concept
If the confidence interval does not contain "0", then there is evidence of a difference in matched pairs.
If the interval contains "0", then there is no evidence of a difference in matched pairs.


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Example

Hank's Auto Mall sells used cars and got them all appraised by Auto Corp. He hired another car appraisal company, Vroom Car Appraisal, to get a second opinion on those same cars. Hank wants to see if there is a significant difference between the appraisal values for each of his cars. Is there evidence of that? Eight cars were randomly sampled. Values are in $,000's:
Auto CorpVroomDifferencex1x2di=x1−x2CAR 15158−7CAR 275669CAR 3876225CAR 4705713CAR 54951−2CAR 6827111CAR 7604218CAR 8604713Meanx‾1=66.75x‾2=56.75d‾=10Standard Deviations1=14.02s2=9.74sd=10.3\begin{array}{|l|c|c|c|}\hline & \text{Auto Corp} & \text{Vroom} & \text{Difference}\\ &x_1 & x_2 & d_i=x_1-x_2\\\hline \text{CAR 1} & 51 & 58 & -7\\\hline \text{CAR 2} & 75 & 66 & 9\\\hline \text{CAR 3} & 87 & 62 & 25\\\hline \text{CAR 4} & 70 & 57 & 13\\\hline \text{CAR 5} & 49 & 51 & -2\\\hline \text{CAR 6} & 82 & 71 & 11\\\hline \text{CAR 7} & 60 & 42 & 18\\\hline \text{CAR 8} & 60 & 47 & 13\\\hline \text{Mean} & \overline{x}_1=66.75 & \overline{x}_2=56.75 & \overline{d}=10\\\hline \text{Standard Deviation} & s_1=14.02 & s_2=9.74 & s_d=10.3\\\hline \end{array}

Construct a 95% confidence interval for the difference in population means.

A 95% confidence interval for the previous example is:

t∗=2.365t^* = 2.365
10±2.365(10.38)10\pm2.365\bigg(\dfrac{10.3}{\sqrt{8}}\bigg)

10±8.61210\pm8.612

[1.388,18.612][1.388, 18.612]
We are 95% confident that the average difference in appraisal value is between $1,388 and $18,612.
It does not contain "0", so there is evidence of a difference in appraisal value.

Practice: Confidence Interval for Matched Pairs

A random sample of 29 students were drawn from a large class. Is there evidence that their performances on the midterm and on the final exam differ, on average? Assume the differences of matched pairs come from a population that is normal.



(i) Construct a 95% confidence interval. Enter your answers below with at least 2 decimal places. (Let the point estimate be negative.)
Extra Practice
Matched Pairs for Two Dependent Means
We wonder if judging differs when a judge give scores during the preliminary competition vs. the finale for talent shows. A random sample of 16 judges reveals the following data:
(a) At the 1% significance level, is there is a significant difference between the prices people are willing to pay based on the posters they are shown?
(b) Explain why we are not doing a two-sample test.
Confidence Intervals for Matched Pairs
Copeman is a private medical clinic. It has a total of 185 locations in B.C. and Ontario. In their efforts to rebrand the company to make it look more luxurious, all the locations were renovated last year. The executives randomly sampled 25 locations from each province and compared their respective sales revenues before and after renovations. Results:
Sample mean of the differences, $5,285
Sample standard deviation of the differences, $25,875
Confidence Interval for Matched Pairs
A 90% confidence interval for the difference in the mean IQ scores between males and females is (−4,7)(-4, 7). Interpret the result.
Confidence intervals for matched pairs
The cabbage diet, invented by Dr.Dumb, aims at helping people lose weight. According to this diet, eating lots of cabbage soup can make you feel full thus reducing the need for unnecessary consumption of calories. Six people are asked to track the number of calories consumed during a regular week and a week when they were on the cabbage diet. The number of calories consumed is recorded below:
Calculate a 99% confidence interval.
Confidence intervals for matched pairs
A matched pairs design hypothesis test is conducted to see if having access to a practice midterm helps improve your test score. The following data is recorded:
A total of 40 students participated in the study.
Calculate a 95% confidence interval.