Interval of Plausible Values

Because the two-sided p-value was not small, we fail to reject the null hypothesis that pi equals 2/3. However, that is very different from saying we have evidence that pi equals 2/3! Keep in mind the mantra that absence of evidence is not evidence of absence! In fact, there are lots of other plausible (believable) values for pi (though 0.74 is not one of them). Can we specify all the believable values? Sure!

We will consider values that are not rejected in a test of significance to be plausible values of pi based on the observed sample proportion. This creates a confidence interval for our parameter.

Basically, we want to see how far we can move our distribution of sample proportions to the left before our observed result is considered in the tail of the distribution, and how far we can move it to the right. Click here for a visualization of this idea.

(i) Use trial-and-error to determine the set of values for pi that would not be rejected if used in the null hypothesis, using the two-sided alternative and 0.05 as the level of significance. [Hints: Use values of pi that are multiples of 0.01 until you can find the boundaries where the exact two-sided p-values change from below 0.05 to above 0.05. Then feel free to “zoom in” to three decimal places of accuracy if you’d like. You might also find the "slider" below the distribution of sample proportions handy.]

Complete the table in your lab report, entering a 0 if the value is plausible and an X if the value is not plausible.


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