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Update inference-paired-means.qmd #504

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8 changes: 4 additions & 4 deletions inference-paired-means.qmd
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Expand Up @@ -436,7 +436,7 @@ Notice that the two observations from the same car are linked with a grey line;
#| across the two tread wear measurements (in cm) from the same car.
#| fig-alt: |
#| Box plots and scatterplot with tire brand on the x-axis and
#| treat on the y-axis. The points are assigned to the x-axis brand
#| tire tread on the y-axis. The points are assigned to the x-axis brand
#| given by the permutation, but the plot differs from previous figures
#| in that it is a second permutation of the brands. Again, the two
#| permuted brands seem equivalent with respect to tire wear.
Expand Down Expand Up @@ -485,7 +485,7 @@ Again, the same observations are linked by a grey line, and some of the tread va
#| across the two tread wear measurements (in cm) from the same car.
#| fig-alt: |
#| Box plots and scatterplot with tire brand on the x-axis and
#| treat on the y-axis. The points are assigned to the x-axis brand
#| tire tread on the y-axis. The points are assigned to the x-axis brand
#| given by the permutation, but the plot differs from previous figures
#| in that it is a second permutation of the brands. The additional
#| permutation demonstrates that the box plots
Expand Down Expand Up @@ -650,7 +650,7 @@ Using the histogram of bootstrapped difference in means, estimate the standard e
The bootstrap SE interval is found by computing the SE of the bootstrapped differences $(SE_{\overline{x}_{diff}} = \$1.64)$ and the normal multiplier of $z^{\star} = 2.58.$ The averaged difference is $\bar{x} = \$3.58.$ The 99% confidence interval is: $\$3.58 \pm 2.58 \times \$ 1.64 = (\$-0.65, \$7.81).$

The confidence intervals seem to indicate that the UCLA bookstore price is, on average, higher than the Amazon price, as the majority of the confidence interval is positive.
However, if the analysis required a strong degree of certainty (e.g., 99% confidence), and the bootstrap SE interval was most appropriate (given a second course in statistics the nuances of the methods can be investigated), the results of which book seller is higher is not well determined (because the bootstrap SE interval overlaps zero).
However, if the analysis required a strong degree of certainty (e.g., 99% confidence), and the bootstrap SE interval was most appropriate (given a second course in statistics the nuances of the methods can be investigated), the results of which bookseller is higher are not well determined (because the bootstrap SE interval overlaps zero).
That is, the 99% bootstrap SE interval gives potential for UCLA bookstore to be lower, on average, than Amazon (because of the possible negative values for the true mean difference in price).

```{r}
Expand Down Expand Up @@ -868,7 +868,7 @@ To compute the test compute the standard error associated with $\bar{x}_{diff}$

$$SE_{\bar{x}_{diff}} = \frac{s_{diff}}{\sqrt{n_{diff}}} = \frac{13.42}{\sqrt{68}} = 1.63$$

The test statistic is the T score of $\bar{x}_{diff}$ under the null hypothess that the true mean difference is 0:
The test statistic is the T score of $\bar{x}_{diff}$ under the null hypothesis that the true mean difference is 0:

$$T = \frac{\bar{x}_{diff} - 0}{SE_{\bar{x}_{diff}}} = \frac{3.58 - 0}{1.63} = 2.20$$

Expand Down