Question

A number of people afflicted with kidney stones have reported that they passed their stones naturally while riding the Big Thunder Mountain Railroad roller coaster at Walt Disney World’s Magic Kingdom theme park. To investigate, Mitchell and Wartinger (2016) brought a silicone pyelocaliceal renal model containing urine and a medium sized kidney stone on the roller coaster ride 20 times. When the model was placed in the front seats, the stone passed once in eight rides. When the model was placed in the rear seats, the stone passed seven times in 12 rides. These data suggest that riding roller coaster is an effective way to pass kidney stones, but does it matter where you sit

a. State the null and alternative hypotheses for a test of whether kidney stone passage rates differ between roller coaster front and back seats
b. Make a table of the expected frequencies under the null hypothesis.

c. Examine the table from (b). What issue arises that would prevent you testing the null hypothesis using the χ2 contingency test?
d. What test method should be used instead?
e. When we carried out the correct test, we obtained P = 0.07, so we failed to reject H0. Which conclusion is more justified: “It does not matter where you sit” or “The study is inconclusive”? Why?

Answer 1

a. The null and alternative hypotheses are:

- Null hypothesis (H0): Kidney stone passage rates do not differ between roller coaster front and back seats.

- Alternative hypothesis (H1): Kidney stone passage rates differ between roller coaster front and back seats.

b. To calculate the expected frequencies under the null hypothesis, you would typically average the total successes (stones passed) over the total trials and apply this proportion to each group based on the number of trials. Given 8 rides in the front with 1 pass and 12 rides in the back with 7 passes, the total is 20 rides with 8 passes.

c. The issue that arises with using the χ2 contingency test here is the small sample size, particularly the low expected frequencies in some cells (especially considering a rule of thumb that all expected frequencies should be 5 or more to use the χ2 test reliably).

d. Given the small expected frequencies in some cells, a Fisher's Exact Test is more appropriate for this scenario as it does not rely on the large sample assumptions that the χ2 test does.

e. Since P = 0.07, which means the probability of observing the data (or more extreme) assuming the null hypothesis is actual is 7%, we failed to reject H0 at the common alpha level of 0.05. The conclusion "The study is inconclusive" is more justified than "It does not matter where you sit" because failing to reject H0 does not prove H0 is true; it only indicates that there is not enough evidence to conclude a significant difference in the effectiveness of kidney stone passage rates between front and back seats based on this study. The p-value close to the threshold suggests that more research could be valuable.