Suppose I want to test whether children are more likely to call heads or tails for a coin flip. I plan to test the hypotheses: Ho: = 0.4 vs. Ha: 0.4. However, my friend plans to use the same sample data to test Ho: = 0.6 vs. Ha: 0.60.

If we each create a null distribution for sample proportions, what will be the main difference between our two distributions?

Question

Suppose I want to test whether children are more likely to call heads or tails for a coin flip.  I plan to test the hypotheses: Ho:  = 0.4 vs. Ha:   0.4.  However, my friend plans to use the same sample data to test Ho:   = 0.6 vs. Ha:  0.60.

If we each create a null distribution for sample proportions, what will be the main difference between our two distributions?

BlackTom AI · Accepted Answer

To begin unpacking the question, we’re comparing two null distributions for sample proportions under different null p-values.
Option: 'The center (mean) of the distributions'. This statement points to the key difference: under a null hypothesis for a proportion p0, the sampling distribution of the sample proportion p-hat has a mean equal to p0. Therefore, if one test uses p0 = 0.4 and the other uses p0 = 0.6, the centers of their null distributions will be at 0.4 and 0.6 respectively, meaning the two distributions are centered at different values.
Next, consider how the spread behaves. For a binomial proportion, the standard deviation of p-hat is sqrt[p0(1 - p0)/n]. Here, p0(1 - p0) equals 0.4×0.6 = 0.24 for both nulls, so the standard deviations are actually the same if the sample size n is the same. Thus, the primary distinguishing feature between the two null distributions is where they are centered, not how spread out they are in this particular scenario.
If we evaluate the other potential differences (even though we don’t have alternative options to compare here), we might note that changing p0 shifts the mean and, in general, can affect skewness and symmetry, but for a fixed n, the variance formula shows identical spread when p0(1 - p0) is the same, which happens to be the case for p0 = 0.4 and p0 = 0.6. This reinforces that the main, explicit difference here is the center.
In summary, the most salient distinction between the two null distributions is indeed the location of their centers, since both have the same spread under these specific p0 values and sample size.

Su25-STATS-141-01/BIO-141-01 3.1, 3.2 & 3.4 Readiness Quiz

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