Question

inferences from experiments (continued)
2 exploration: evaluating results
work as a class. to conclude that the treatment is responsible for the difference in yield, you need strong evidence to reject the hypothesis:
water dissolved in calcium has no effect on the yields of yellow squash plants.
to evaluate this hypothesis, compare the experimental difference of means with the resampling differences.
a. collect all the resampling differences of means found in exploration 1(b) for the whole class and display these values in a histogram.
b. draw a vertical line on your class histogram to represent the experimental difference of means found in exploration 1(a).
c. where on the histogram should the experimental difference of means lie to give evidence for rejecting the hypothesis?
d. is your class able to reject the hypothesis? explain your reasoning.

Explanation:

This problem involves evaluating experimental results, specifically about the effect of water dissolved in calcium on yellow squash yields, using resampling and hypothesis testing. The subfield under Natural Science that best applies is Biology (since it deals with plant growth/yields) or Statistics (for the resampling and hypothesis testing methods). Given the experimental design and data analysis (histograms, difference of means), Statistics (under Mathematics) or Biology (under Natural Science) could apply, but since it's an experiment on plants, Biology is more context - specific for the subject matter, while Statistics is for the analysis method. For the question about where the experimental difference of means should lie to reject the hypothesis: In hypothesis testing (especially with resampling), if the null hypothesis is that the treatment (water with calcium) has no effect, we expect that under the null, the difference of means from resampling (simulating no treatment effect) would be centered around 0 (or a value indicating no difference). To reject the null, the experimental difference of means should lie in the tail(s) of the resampling distribution (the histogram of resampling differences), i.e., in the extreme regions where such a difference is unlikely to occur by chance if the null were true. For example, if the resampling distribution is symmetric around 0, the experimental difference should be far from 0 (in the upper or lower tail, depending on the direction of the effect) such that it's in the extreme percentiles (like the 5% or 1% tails, depending on the significance level).

Answer:

To reject the hypothesis that water dissolved in calcium has no effect on the yields of yellow squash plants, the experimental difference of means should lie in the extreme tail(s) of the histogram of resampling differences of means. Specifically, it should lie in a region where the proportion of resampling differences that are as extreme or more extreme is small (e.g., in the upper or lower 5% or 1% of the resampling distribution, depending on the chosen significance level). This is because if the null hypothesis were true (no effect of the treatment), we would expect the experimental difference to be similar to the resampling differences (which simulate the null). If it's in the extreme tail, it suggests that the observed difference is unlikely to have occurred by chance under the null, providing evidence to reject it.