Testing a probability table for distribution properties

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to test a probability table for distribution properties.  Here's our problem statement: Groups of adults are randomly selected and arranged in groups of three.  The random variable x is the number in the group who say they would feel comfortable in a self driving vehicle.  Determine whether a probability distribution is given.  If a probability distribution has given, find its mean and standard deviation.  If a probability distribution is not given, identify the requirements that are not satisfied.

Part 1

OK, so this first part of the problem is asking us whether we have a probability distribution listed in the table.  In order for these, uh, this distribution table to qualify as a probability distribution, three requirements need to be met.  First is that we're actually listing actual probability values.  And to test that, we look at each of the values that are listed here in the probability column.  And the most important property of the probability that we're going to be looking for here is that it needs to be a value between 0 and 1, inclusive.  So we look at each one of these values here, and they're all between 0 and 1.  So therefore the first requirement is met.

The second requirement is that there must be a matched value for each possible random variable.  So if we look at our random variable, which in this case is the x, we have a probability value listed for each one of those random variable values.  So the second requirement is met.

The third requirement is that each of these probabilities listed in the probability column must add up to equal 1.  And by 1, I mean 1 exactly.  So you're not getting like 0.99 or 1.01; it's gotta be 1 exactly.  So there's two ways to do this.  I find it easier to do this in Excel.  So I'm just going to dump the data here in Excel.  And it always brings up whatever other copy of Excel I have open.  When it does this, I'm just gonna make this easy.  We're just going to paste you.  So notice here when I actually have the data here in Excel, if I just select these columns with the probability, down here at sums it for me automatically.  This is why I say it's easiest to do this in Excel.  All I have to do is dump the data in, then get the column selected, and then here I get the sum, and it's 1 exactly.  So the third requirement is met.

Now if I wanted to do this in StatCrunch --- StatCrunch sometimes gets a little clunky.  This particular operation is not that bad.  I can actually put my window here and see what's going on.  So here in StatCrunch, I'm going to go to Stat --> Summary Stats --> Columns and then sum up that probability column.  We see we get the same thing.  It's equal to 1 exactly.  So we do have a probability distribution here.  So I'm going to select that answer here from among my choices.  Well done!

Part 2

And now the second part of this problem asks me to find the mean of the random variable x.  This is not where you're just adding up 0, 1, 2, 3 and then dividing by 4.  That's not going to give it to you.  What will give it to you is when you actually calculate the --- what it's asking for is the mean of the distribution.  And so that's easiest to do in StatCrunch.

The way we're going to do this, just go to Stat --> Calculators --> Custom.  In the options window, our values are going to be our random variable, which in this case is the x, and the weights are going to be our probabilities.  I press Compute!, and out comes this wonderful little results window here with the mean listed right here for you.  It's already calculated for you.  So I'm going to go ahead and stick that in my answer field.  It says to round to one decimal place.  Nice work!

The last part of his problem is asking us for standard deviation.  And again that's listed right over here in our results window.  So I'm going to go ahead and stick that value here in the answer field.  Nice work!

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