Using StatCrunch to construct a confidence interval for two matched pair means of hospital admissions
Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to use StatCrunch to construct a confidence interval for two matched pair means of hospital admissions. Here's our problem statement: Data on the numbers of hospital admissions resulting from motor vehicle crashes are given below for Fridays on the 6th of a month and Fridays on the following 13th of the same month. Assume that the paired sample data is a simple random sample and that the differences have a distribution that is approximately normal. Construct a 95% confidence interval estimate of the mean of the population of differences between hospital admissions. Use the confidence interval to test the claim that, when the 13th day of a month falls on a Friday, the numbers of hospital admissions for motor vehicle crashes are not affected.
OK, here we have our data, and here we have the first part of our problem asking us to construct a confidence interval. To do this, I'm going to dump the data into StatCrunch. I'm going to come here and resize my window so we can see everything. Great!
Now I'm going to construct my confidence interval on the data that was given. To do that, I go up to Stat –> T Stats –> Paired. Notice there's no extra menu option here with the parent option. That's because StatCrunch was coded under the assumption that if you want to run hypothesis testing or construct a confidence interval with matching pairs in your data set that you already have actual data to supply. So there's no summary option here.
Here in my options window, I found it's best practice when identifying between two samples in the options window that the first sample should always be the first grouping or the grouping that's mentioned first in the problem statement (in this case, that's Friday the 6th). And then Sample 2 will be the second grouping or the grouping mentioned second (in this case, Friday the 13th). You're asked to construct a confidence interval, so I'm going to click the radio button here on confidence interval.
And normally in these types of problems we’re given a significance level, or alpha, and then it's assumed then that we'll use that information to compute or calculate the confidence level that is appropriate for a confidence interval. However, in this case we're actually given a confidence level; 95% is specified here in the problem statement. So I'm just going to use that specified confidence level for my interval. That's the default selection, so I don't need to make any changes there. I press Compute!, and here is my results window.
If I scroll over here, at the end of the table I can see the lower and upper limits that I'll need to use for my confidence interval. So I'm going to put those in here. Good job!
Now the second part of the problem asks us to interpret the confidence interval with respect to the claim. The value that we want to check for in our confidence interval is going to be zero. If there's no difference between the two samples, or between the sample means rather, then zero should be inside the confidence interval. And if zero is inside the confidence interval, that means it's potentially the value of the mean difference between the two means. If the difference between the means is zero, that means that the means of that are the same, and so there's no real difference between the two groupings.
However, if zero is not inside the confidence interval, then that means there is some difference between the groupings. And that means one of those groupings is going to have a greater mean value than the other. And which one of those two groupings it is depends on which side of the confidence interval zero appears, whether it's to the left to the right. Here we're simply asked about the claim.
So we check to see if zero is inside our confidence interval. Here's our confidence interval, and zero is not inside the confidence interval. That means there is going to be some difference between the means. So the claim is that just because, you know, the hospital admissions falls on the 13th day of a month which is a Friday and that therefore the motor vehicle crashes are not affected. Well, the number of hospital admissions is affected because there's some difference there between the mean values. If they weren't affected, then that would mean that the mean value difference would be zero. But it's not zero, because zero is not inside the confidence interval. Therefore, there is some difference, and so we we can't actually reject the claim that they're not affected because we have statistical evidence that they are affected.
So among the answer options given here, I want to select this one: Yes, we can reject the claim because zero is not included in the confidence interval. We have statistical evidence that suggests there is some difference here between the mean values for the groupings. Well done!
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