This is the problem I was assigned in my homework:

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I read in the textbook and believe that this method should be applied to solve the problem when dealing with 2 independent variables.

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As a result, I've done so hand-written work and filled in the formula like this:

var(aX + bY) = 48^2 (1) + 6^2 (.0625)

However, as I have found incredibly weird and most likely wrong, the variance came out to be 2306.25, and thus the calculated standard deviation was 48.02. This seems wrong to me, but I can't lay my finger on it.

Explanation of variables and why I chose this formula

I'm on part A, and I calculated the expected amount of ice cream to be served in total to be 48 + 3 scoops of a mean of 2 oz each = 54 oz.

I went on to the second part of A, and came up with the formula above. Is there something wrong with how I set it up?

Edit: On Part B now

Expected value of ice cream being left in the box is: 48 - 2 = 46 oz. Variance formula: var(aX + bY) = 1^2 * 1 + 1^2 * .0625 = 1.0625 = 1.03


For the first part of (a), you probably should have uses $a=1$ ("one box") and b=3 ("plus three scoops from a second box") in the combined variance formula. But if the scoops are independent of each other and the box then you should be using a formula more like $Var(X+Y_1+Y_2+Y_3) = Var(X)+Var(Y_1)+Var(Y_2)+Var(Y_3)$ so here $1+0.0625+0.0625+0.0625$ [edited - see OpalE's comment]

You will get a more credible standard deviation.

  • $\begingroup$ Which works out to be 1.25. $\endgroup$ – Parcly Taxel Aug 12 '16 at 0:30
  • $\begingroup$ @Henry - Then the formula looks like this: var(aX + bY) = 1^2 * 1 + 3^2 * .0625, which equals 1.5625? $\endgroup$ – Jacob Macallan Aug 12 '16 at 0:33
  • $\begingroup$ @Xari Yes: though you need the square root for a standard deviation $\endgroup$ – Henry Aug 12 '16 at 0:36
  • $\begingroup$ Thank you! And this standard deviation is the amount of ice cream being served, right? In what units though? I'm having a bit of trouble wrapping my head around the fact that I just used a unit of box with units of scoops to get a combined variance. Is the SD calculated from this ice cream in oz, or just ice cream in total? @Henry $\endgroup$ – Jacob Macallan Aug 12 '16 at 0:40
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    $\begingroup$ I'm concerned that since each scoop is itself randomly sampled, you might possibly be needing to use more than 2 independent variables: I think you should have 3 independent scoops represented as Y1, Y2, and Y3. Refer to this AP Statistics problem from several years back: media.collegeboard.com/digitalServices/pdf/ap/apcentral/… (part ii). This would result in you NOT squaring the 3. $\endgroup$ – Opal E Aug 12 '16 at 0:53

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