7.2: Paired data
- Page ID
- 56945
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In an earlier edition of this textbook, we found that Amazon prices were, on average, lower than those of the UCLA Bookstore for UCLA courses in 2010. It’s been several years, and many stores have adapted to the online market, so we wondered, how is the UCLA Bookstore doing today?
We sampled 201 UCLA courses. Of those, 68 required books could be found on Amazon. A portion of the data set from these courses is shown in Figure [textbooksDF], where prices are in US dollars.
| subject | coursenumber | bookstore | amazon | pricedifference | |
|---|---|---|---|---|---|
| 1 | American Indian Studies | M10 | 47.97 | 47.45 | 0.52 |
| 2 | Anthropology | 2 | 14.26 | 13.55 | 0.71 |
| 3 | Arts and Architecture | 10 | 13.50 | 12.53 | 0.97 |
| \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) |
| 68 | Jewish Studies | M10 | 35.96 | 32.40 | 3.56 |
Paired observations
Each textbook has two corresponding prices in the data set: one for the UCLA Bookstore and one for Amazon. When two sets of observations have this special correspondence, they are said to be .
Paired data Two sets of observations are paired if each observation in one set has a special correspondence or connection with exactly one observation in the other data set.
To analyze paired data, it is often useful to look at the difference in outcomes of each pair of observations. In the textbook data, we look at the differences in prices, which is represented as the variable in the data set. Here the differences are taken as
\[\begin{aligned} \text{UCLA Bookstore price} - \text{Amazon price}\end{aligned}\]
It is important that we always subtract using a consistent order; here Amazon prices are always subtracted from UCLA prices. The first difference shown in Figure [textbooksDF] is computed as \(47.97 - 47.45 = 0.52\). Similarly, the second difference is computed as \(14.26 - 13.55 = 0.71\), and the third is \(13.50 - 12.53 = 0.97\). A histogram of the differences is shown in Figure [diffInTextbookPricesF18]. Using differences between paired observations is a common and useful way to analyze paired data.
Inference for paired data
To analyze a paired data set, we simply analyze the differences. We can use the same \(t\)-distribution techniques we applied in Section 1.
| \(n_{_{\text{\emph{diff}}}}\) | \(\bar{x}_{_{\text{\emph{diff}}}}\) | \(s_{_{\text{\emph{diff}}}}\) | ||
| 68 | 3.58 | 13.42 |
Set up a hypothesis test to determine whether, on average, there is a difference between Amazon’s price for a book and the UCLA bookstore’s price. Also, check the conditions for whether we can move forward with the test using the \(t\)-distribution. [htSetupTextbookPriceDiff] We are considering two scenarios: there is no difference or there is some difference in average prices.
- \(\mu_{\text{\emph{diff}}} = 0\). There is no difference in the average textbook price.
- \(\mu_{\text{\emph{diff}}} \neq 0\). There is a difference in average prices.
Next, we check the independence and normality conditions. The observations are based on a simple random sample, so independence is reasonable. While there are some outliers, \(n = 68{}\) and none of the outliers are particularly extreme, so the normality of \(\bar{x}\) is satisfied. With these conditions satisfied, we can move forward with the \(t\)-distribution.
Complete the hypothesis test started in Example [htSetupTextbookPriceDiff]. [SEAndTScoreTextbookPriceDiff] To compute the test compute the standard error associated with \(\bar{x}_{\text{\emph{diff}}}\) using the standard deviation of the differences (\(s_{_{\text{\emph{diff}}}} = 13.42{}\)) and the number of differences (\(n_{_{\text{\emph{diff}}}} = 68{}\)):
\[\begin{aligned} SE_{\bar{x}_{\text{\emph{diff}}}} = \frac{s_{\text{\emph{diff}}}}{\sqrt{n_{\text{\emph{diff}}}}} = \frac{13.42{}}{\sqrt{68{}}} = 1.63{} \end{aligned}\]
The test statistic is the T-score of \(\bar{x}_{\text{\emph{diff}}}\) under the null condition that the actual mean difference is 0:
\[\begin{aligned} T = \frac{\bar{x}_{\text{\emph{diff}}} - 0} {SE_{\bar{x}_{\text{\emph{diff}}}}} = \frac{3.58{} - 0}{1.63{}} = 2.20 \end{aligned}\]
To visualize the p-value, the sampling distribution of \(\bar{x}_{\text{\emph{diff}}}\) is drawn as though \(H_0\) is true, and the p-value is represented by the two shaded tails:
The degrees of freedom is \(df = 68{} - 1 = 67{}\). Using statistical software, we find the one-tail area of 0.0156. Doubling this area gives the p-value: 0.0312.
Because the p-value is less than 0.05, we reject the null hypothesis. Amazon prices are, on average, lower than the UCLA Bookstore prices for UCLA courses.
Create a 95% confidence interval for the average price difference between books at the UCLA bookstore and books on Amazon.
We have strong evidence that Amazon is, on average, less expensive. How should this conclusion affect UCLA student buying habits? Should UCLA students always buy their books on Amazon?