8. Testing for Differences in Means and Proportions: Paired Samples t-test
Confidence Interval for a Mean Difference
A researcher conducts an experiment in which #12# randomly selected students are invited to eat dinner at a restaurant on two different evenings. On one evening each student receives a regular-size plate and on the other they receive a large-size plate.
On each occasion, the students are allowed to choose as much food as they want from a buffet. Once the students have made their selection, their plates are weighed.
The table below shows how much food (in grams) each student chose when they were a given a regular-size plate #(X)# and when they were given a large-size plate #(Y)#:
| Student | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| #X:\,\text{Regular}# | 363 | 304 | 337 | 390 | 357 | 343 | 402 | 406 | 390 | 300 | 385 | 346 |
| #Y:\,\text{Large}# | 319 | 284 | 351 | 408 | 462 | 372 | 446 | 393 | 411 | 322 | 408 | 417 |
You may assume that the amount of food eaten for either plate size is normally distributed.
Define #D=Y-X# and construct a #90\%# confidence interval for the population mean difference #\mu_D#. Round your answers to #3# decimal places.
#CI_{\mu_D,\,90\%}=\,(##,\,\,\, ##)#
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