If you are given the equation of a line and you need to plot it, use this **WolframAlpha** calculator: https://www.wolframalpha.com/input/?i=line,+slope%3D2.75,+y-intercept%3D.5

I’m sure most of us are familiar with Frost’s famous poem, *The Road Not Taken*. I was reminded of the last two lines this morning by an article in Inc. on Jeff Bezos. It makes the point that most often in life we don’t regret the things we do. Rather, we regret the things we didn’t do.

When considering a major life decision, those of us who like numbers, can “quant” it to many decimal places.

But Bezos says that those life decisions are times when we should listen to our heart, perhaps more than to the logical reasons not to do something, though I think a bit of quant is good. The article suggests we should look for the greater meaning in those decisions such as “creating a better future for your family, wanting to make a difference, or hoping for a more rewarding and fulfilling life.”

Though we may not succeed all the time, we will be less likely to have the regret of not having tried at all.

“Two roads diverged in a wood, and I – I took the one less traveled by, and that has made all the difference.”

Haden, J. (2018, Jan 8). *Jeff Bezos: Ask Yourself 1 Question to Make Truly Important Decisions (and Avoid Lifelong Regrets)*. Retrieved from Inc.: https://www.inc.com/jeff-haden/jeff-bezos-ask-yourself-1-question-to-make-truly-important-decisions-and-avoid-lifelong-regrets.html

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Procrastination. I think we all suffer from this human problem. And we all probably have a sense that it rarely leads to optimum outcomes, though the ancient Greeks and Romans thought it was preferable to do things later after more reflection. (Partnoy, 2012)

Studies of the impact of procrastination on outcomes in online courses tell a different story. Waiting until the end of the window to complete quizzes and exams leads to lower grades. (Cerezo, Esteban, Sanchez-Santillan, & Nunex, 2017) (Levy & Ramin, 2012)

It is as simple as that.

Some of this is due to the “cramming” effect our teachers warned us about all through our schooling – not really having time to adsorb the exam material. Some is possibly due to just running out of time to complete the assignments – students kicked out of an exam at midnight on the due date. But waiting until late in the week to do the work will likely lower your grade.

One reason many students choose online education is the freedom to choose when and where to study. And work and family life does happen while in an online course, restricting our ability to study. But where you can, plan to work on assignments and quizzes/exams as early in the week as you can. It will pay off in higher grades.

Cerezo, R., Esteban, M., Sanchez-Santillan, M., & Nunex, J. (2017). Procrastinating Behavior in Computer-Based Learning Environments to Predict Performance: A Case Study in Moodle. *Front. Psychol*, 8: 1-11.

Levy, Y., & Ramin, M. (2012). Procrastination in Online Exams: What Data Analytics Can Tell Us. *Proceedings of the Chais conference on instructional technologies research 2012: Learning in the technological era* (pp. 41-49). The Open University of Israel.

Partnoy, f. (2012). *Wait: The art and science of delay.* New York: Public Affairs.

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This question comes up frequently in my intro stats class. We are covering paired (dependent) samples hypothesis tests and the explanation in the textbook gives the students a fuzzy discussion and includes a complex formula for finding the latter.

“d-bar” (my editor is not letting me use the correct symbol, but this is the letter “d” with a bar over it) is just the average of the differences (d) in the two samples.

“s sub d” (editor again) is just the plain old standard deviation of the differences, despite this rather complex formula:

Here is an example of how to find them using basic Excel functions:

Here is the Excel worksheet with formulas:

Unfortunately, StatCrunch does not give you “s sub d” directly in any of the built-in analyses and but you can find them both by the following method:

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The general rule is to not round any intermediate values in a series of calculations and, instead, carry forward as many decimal places as possible. For example, if you need to first calculate the standard deviation for use in a later equation, keep as many decimal places as possible (no less than 6 is my recommendation) in your standard deviation value.

But when you enter your answers into MyStatLab (MSL), round to the required number of decimal places. When MSL says “two decimal places as required” or “one decimal place as needed,” that means **exactly** that many of decimal places are required.

In this example from 7.3, MSL asks for the “nearest thousandth as needed.” That means you must enter exactly three numbers after the decimal.

If you enter 2.85, you will be counted wrong. If you enter 2.8449, you will be counted wrong. For example, see images below:

Generally, MSL does allow a tolerance level around the exact answer, e.g. 4.2004 + or – 0.0001. And on some problems, MSL also includes alternate answers where “technology” gives a slightly different value than interpolating from a table, again within a tolerance range.

But when it comes to the number of decimal places, MSL is strict.

This may seem a bit picky, but in the real world, our bosses will not be happy if we do not follow instructions.

]]>Later in the course, things will get more complicated re degrees of freedom.

Let me try to give a short explanation of degrees of freedom for this part of the course.

Suppose we have a sample of five weights of puppies. To the nearest pound – we have a cruddy scale – they are 3, 4, 4, 5, ?. If I tell you the total of the five weights is 22 pounds, and then tell you the first four weights (3, 4, 4, 5), can you find the 5th weight? You should say “Sure!” Just add up the four weights – 16 pounds – and subtract from the overall total – 22 pounds – to find the missing weight is 6 pounds.

Once you know the four of the five, the fifth weight is not free to vary. That sample of 5 puppy weights has just 4 degrees of freedom, (n-1).

Although the math gets tricky in later parts of the course (after the midterm), the concept of degrees of freedom is similar to this.

]]>[My **Excel** calculator for running this test is found here.]

As always for proportion problems, we have to check first to be sure n*p and n*q or n*(1-p) are both > 5.

If they are, we can use the normal approximation to the binomial (a proportion is essentially a binomial test). That is why f**or all the proportion problems we do in this course we use the z-test**. If either n*p or n*q is not > 5, we have to use a test that is beyond the scope of this course.

Step 1 as almost always is true for us, is to click on **Stat** once StatCrunch is open. The sequence is **Stat>Proportion Stats>One Sample>With Summary**. Then enter the number of success, which is just n*p, the sample size, select the Hypothesis test for p, and enter the null hypothesis and select the math operator for Ha. Click **Compute!** The answer window opens and you can see the test statistic of z= -1.042 and the p-value of 0.29. Although this problem asks us to find the critical value and make our decision based on that, the p-value always agrees and tells us we Fail to Reject the null since p>α.

I used the **Stat>Calculators>Normal** path to bring up the normal calculator. Because this is a two-tailed test (recall Ha has the ≠ math operator), I like to use the **Between** button and enter the confidence level c, which is 1-α or 0.9. The calculator shows the two critical values and rejection areas. Because the test statistic of -1.042 does not fall in either rejection area, we get the same decision of **Fail to Reject the null**.

The final part is to draw a conclusion. Recall the alternative was the claim, so we say: **Fail to Reject Ho. The data do NOT provide sufficient evidence to Support the claim.**

Boys will be boys: Data error prompts U-turn on study of sex differences in school (Retraction Watch, 2017)

The article is about a peer-reviewed article on self-regulation of study habits that was published earlier this year. In the retraction, the authors noted they had discovered a “coding error” that flipped the outcome of their research. Although the standard procedure for using coding dummy variables is “1” = Yes and “0” = No, that procedure trips up a bit when it comes to gender. Generally, male is coded “1” but here the researcher doing the basic coding used “1” to mean female. No one noticed and they drew their conclusions as if the data had been coded the opposite way with “1” meaning male.

I noticed several students made a similar “error” when setting up the multiple regression in M3A2 on the value of a fireplace. And that was a “coding error” when using a dummy variable to include the categorical variable “fireplace” in the regression.

As you probably know, a regression requires quantitative variables but, in our database, we have a variable Fireplace that contained either a True or False text value. When we use a dummy variable, we replace a categorical text value with a number. In the Fireplace problem, a logical way to do that, logical for me, would be to use a “1” to indicate the presence of a fireplace and a “0” to indicate no fireplace. Some students chose to do the reverse, “0” for Fireplace = True and “1” for Fireplace = False.

The error I am speaking of is not that – deciding to use “0” for Fireplace = True. The error comes in not understanding what either choice means for how you interpret the outcome of the regression.

If you chose “0” for Fireplace = True and did the multiple regression correctly, you came out with a beta2 coefficient of about -$5567 for the dummy variable.

If you did the opposite and used “1” for Fireplace = True, you found a beta2 coefficient of +$5567.

The error happens when you try to interpret these outcomes.

It is straightforward to interpret the outcome if you let “1” indicate the presence of a fireplace, Image 1. That is that having a fireplace adds about $5567 in value to the sales price of a typical house. You see this when you use the CI/PI worksheet to forecast home values by putting in either a “0” or a “1” in the calculator.

Image 1

But some students who used “0” to indicate the *presence of a fireplace* came up with an incorrect conclusion: that the *presence of a fireplace reduced* the price of a home because beta2 was negative.

In reality, for the students who used “0” to indicate the presence of a fireplace, the negative beta2 tells you that ** not** having a fireplace reduces the value of a home, just the opposite.

Image 2

Final thought: you may notice that the y-intercepts on the two coding methods are also different. But if you check, you will see that the y-intercept for Fireplace=True=0 is $5567 greater than the y-intercept of Fireplace=True=1. That makes sense because under that scenario, the starting point should be greater because the assumption is that the house has a fireplace.

Retraction Watch. (2017, Oct). *Boys will be boys: Data error prompts U-turn on study of sex differences in school*. Retrieved from Retraction Watch: http://retractionwatch.com/2017/10/17/boys-will-boys-data-error-prompts-u-turn-study-sex-differences-school/

Remember since these are independent events – the probability of choosing 4 from shift 1 does not change for different choices from shift 2 – the P(A and B) = P(A)*P(B) using the Multiplication Rule.

Download a copy of the Excel workbook here.

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