Paired Data: Estimating a Mean Difference
(Lesson 10.5)
Chapter 10 - Day 7
Chapter 10
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8
Day 9
Day 10
Day 11
Day 12
Day 13
Day 14
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Learning Targets
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Use a graph to analyze the distribution of differences in a set of paired data.
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Calculate the mean and standard deviation of the differences in a set of paired data and interpret the mean difference.
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Use the four-step process to construct and interpret a confidence interval for a mean difference.
Experience First
Today's lesson involves a very real and relevant topic: climate change. The lesson was co-developed along with Dash from Skew The Script. Be sure to save your data from this lesson as it will also be used for the next Lesson 10.6.
We suggest you model once for students how they will be collecting data. Here are the instructions (which are also included in the student handout).
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Data will be collected from a NASA database that can be accessed from www.tinyurl.com/NASA-Globe.
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Start with a random number between -180 and 180 (longitude) and a random number between -90 and 90 (longitude). Use random.org or RandInt on the calculator to get these values.
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Set the dates above the globe to be “from January 1900” and “to January 2000.” Choose the “GHCN V4 adj-cleaned” dataset. Click the “Update” button.
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Below the globe, type in your randomly selected latitude and longitude. Then click “Recenter.” Zoom in by double clicking on the globe until only one station appears in your window (the nearest station to your chosen latitude and longitude).
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Hover over your chosen station and click “generate plot.” Using the graph, find the highest point between 1900-1940 and estimate its temperature (in Celsius). Put your estimate in the “Temp. 1” space below. Do the same thing for the years 1960-2000. Round your temperature estimates to one decimal point. Then, find the difference.
Each student should randomly select one location on the globe and find the difference in temperature. They should then record the difference on the board.
The 4-step confidence interval to estimate the mean difference in temperature is exactly the same as the one sample t interval for a mean that students learned back in Lesson 8.6 (but this time using differences as the data set!). Students will use the One sample t interval for a mean applet (part of the Traditional Inference applets at www.statsmedic.com/applets) for the calculation of the interval.
Formalize Later
When debriefing the activity, here are a few items to highlight:
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When defining the parameter, it is important to include the words "mean difference" or "mean of the differences".
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If the confidence interval contains 0, it is plausible that there is no difference in the temperature from the beginning to the end of the 1900s. If both endpoints of the interval are positive, then we have convincing evidence of global warming.
Notice that "mean difference" is not the same as "difference of means". Estimating a mean difference (Lesson 10.5) is a one sample procedure. Estimating a difference of means (Lesson 10.3) is a two sample procedure.