Refer to Data Set 3 "Body Temperatures" in Appendix B and...

Refer to Data Set 3 "Body Temperatures" in Appendix B and use all of the matched pairs of body temperatures at 8 AM and 12 AM on Day 1 . When using a $0.05$ significance level for testing a claim of a difference between the temperatures at $8 \mathrm{AM}$ and at $12 \mathrm{AM}$ on Day 1 , how are the hypothesis test results and confidence interval results affected if the temperatures are converted from degrees Fahrenheit to degrees Celsius? What is the relationship between the confidence interval limits for the body temperatures in degrees Fahrenheit and the confidence interval limits for the body temperatures in degrees Celsius? Hint: $C=\frac{5}{9}(F-32)$.

Key Concepts

Effect on Hypothesis Testing

When the data are subjected to a linear transformation, the resulting test statistics (such as t-scores) remain unaffected in their significance because the transformation scales both the mean difference and the standard error proportionally. This means that the outcome of the hypothesis test (rejecting or failing to reject the null hypothesis) is invariant under the conversion.

Confidence Interval Adjustment

Under a linear transformation, the confidence interval limits for a parameter, such as the difference in means, are transformed using the same linear formula. This has the effect of converting each limit of the interval in the original units (Fahrenheit) to the new units (Celsius) without altering the confidence level. The endpoints of the confidence interval in Celsius are exactly obtained by applying the transformation to the endpoints determined in Fahrenheit.

Matched Pairs Design

This design involves collecting paired measurements from the same subject under different conditions or times, allowing for the direct analysis of differences within each pair. It is useful to control for subject-specific variability and is especially relevant when comparing measurements such as body temperatures taken at two different times.

Linear Transformation

A linear transformation, such as converting Fahrenheit to Celsius with the formula C = 5/9(F - 32), involves scaling and translating data. In the context of hypothesis testing and confidence intervals, such transformations preserve the structure of differences because they are applied uniformly to all measurements, meaning that the statistical conclusions remain equivalent under the transformation.

Transcript

00:01 Okay, so for this one, we're redoing the body temperature data set problem, where we test the body temperature at 8 a .m.

00:09 And 12 a .m.

00:11 The same day.

00:12 And this time, we're testing them with the given values in fahrenheit, and then we're converting in the celsius, and bring them again to see if there's any difference.

00:23 So i've written the null and alternate hypotheses here in different colors, just showed that they're technically different, even though basically the the same but i've imported the data in the mini tab so let me pull it up okay so here i have the data set loaded you can see in columns 8 and 9 here i've already converted the values into celsius i'll show you how to do it and i click on column 11 here i right click go to formulas assign formula to column there's this expression window so now i have to enter the formula to convert fahrenheit to celsius i go added by 9 times parentheses.

01:05 And here i have to click the column that i'm converting.

01:12 32.

01:14 That's the formula.

01:15 So it's going to fill the column.

01:17 Everything from column 3.

01:21 Converted to celsius.

01:22 I hit ok.

01:27 Write that also as d1 .8c.

01:31 Just to make sure i don't get confused.

01:34 Used and it won't let me so i'll write that to do the same thing with the 12 a .m.

01:43 Values.

01:43 I'll load formulas, five divided by nine, these, m minus 32.

02:03 So i'll write d1 20, i guess zero, am c2.

02:13 Now we just want to check the confidence center rule and the hypothesis tests for these in celsius and see if there are any different ones in fahrenheit.

02:29 So i'll start with the ones in fahrenheit that they gave us, columns three and four.

02:33 I'll go up here to stat.

02:36 Basic statistics...

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