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In Class Quiz: An Excel – Based Interpretation of Confidence

The purpose of this quiz is to get at the true meaning of “confidence” concretely (statistical confidence, that is; I figure everyone has their own image of what “nonmathematical confidence” looks like in the world around them). We’re used to that word “confidence”. Goodness knows we see it enough – here are two instances I noticed “95% confidence” in just this week!

|[pic] |[pic] |

But what, exactly does “95% confident” mean? That’s what this quiz aims to do: shed light on the idea of confidence in a very concrete way. We’ll use Excel to help!

In this quiz, we’ll approximate the average birth weight of an American baby (ABWAB), using the ideas of confidence intervals (CIs). Please start by opening the spreadsheet “weights.xlxs” (the link next to the link from which you opened this page). You’re viewing the ABWABs of 20 randomly selected American newborns[1], along with a histogram of that sample. Let’s start by getting that mean and standard deviation filled in! Type =AVERAGE(B4:U4) in cell D8 and =STDEV(B4:U4) in cell D9. Don’t forget the equals signs!

Now – press [pic] key a few times. Each time, you get a new “sample of 20 babies”. You see how their weights change each time? The histogram, and s all change, as well.

From these data, I’d like you to construct some 95% CIs for the ABWAB, using Excel. We’ll use the “confidence” command to get the Margin of Error (MOE) of our sample of size 20. In cell D11, type this in:

[pic]

• The first value, 0.05 is called the significance. It’s the decimal complement of the confidence. Since the significance is set at 5% the confidence must be 95%.

• The second value is a cell reference to the sample standard deviation, which you placed in cell D9.

• The third number is the sample size.

| | |

Excel only needs these three values for a MOE calculation; we’ll see more later in class!

Now that you have a mean and a MOE, you can construct the CI’s! Tech note: the two endpoints of any confidence interval are often called the “LCL” (“lower confidence limit”) and “UCL” (“upper confidence limit”). Let’s form those now!

Type =D8-D11 in cell C15, and =D8+D11 in cell D15. Cells C15 and D15 are now your 95% CI endpoints! Cool, huh?

Go ahead and press [pic] a few more times and see how everything changes. When you’re bored of doing that, I want you to gather some 95% CI’s!

1. (8 points) Give me twenty 95% CI’s for the ABWAB. Press [pic] at least once after constructing each CI so that you get new baby data each time. Round values to the nearest hundredth’s place. You can most likely just copy and paste them into a Word document.

OK...so you have 20 CI’s. Great! Now, we need to drive home what the term “95% confident” really means...

…it is “known” that the ABWAB is 7 pounds, 11 ounces (or 7.67 pounds). Take a look at your CI’s:

2. (2 points) What percent of the CI’s that you constructed contain the value ( = 7.67?

(wait’ll you see what happens when we collate our data. ( )

| It gets better...please click on the tab “Randomized |[pic] |

Results” (it’s down near the bottom of the workbook). Here, I’ve created 20 CI’s (identical, in construction, to the 20 you just created). However, instead of giving you the values of the endpoints, I’ve done the CI’s graphically.

You’ll also see a vertical line through 7.67; this line represents the value of (. Do you see how many of

|These CI’s contain (? I did this little exercise, too, and got the results at right. Check this out: |Percent that contain ( |

| |f |

|The mode of these values? 95%. | |

| |80 |

|The median of these values? 95%. |1 |

| | |

|The mean of these values? A little above 95%. |90 |

| |3 |

| | |

| |95 |

| |10 |

| | |

| |100 |

| |8 |

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Wowsers! This means that, if we repeatedly construct CI’s around randomly selected data sets, we can expect around 95% of them to contain the true population mean.

And that, my fine friends, is the definition of confidence!

Make sure you submit your 20 CI’s (from question number 1) and your percent (from question number 2)!

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[1] Actually, what you’re looking at is a sample of 20 numbers I generated using the “known” values of ( and ( for the weights of newborns. By “known”, I mean that, after hundreds of millions of births, doctors have pretty much figured that they have the parameter values of ( and (. I used Excel’s random number generator (and something called the Box – Muller normal approximation) to allow for natural fluctuations from baby to baby; every time you press ENTER, the randomizer will kick in, and you’ll get 20 new baby weights. If you’re interested, click in any of the weight cells to see the function.

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