Use frequency distribution formulas to estimate the sample mea...

Question

Use frequency distribution formulas to estimate the sample mean and the sample standard deviation of the data set in Exercise 2.

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says using the class midpoints and corresponding frequencies estimate the sample mean and standard deviation of the provided data, and we're giving our data in the table. And in our table, in the first column, we have our class intervals, and we have the intervals of 10 to 1920 to 29, 30 to 39, 40 to 49, and 50 to 59. And in the second column of our table, we have our frequencies, and we have the values of 47, 10, 6, and 3. We're also given 4 possible choices as our answers. For choice A, we have X bar is equal to 33.5, and S is equal to 13.2. For choice B, we have X bar is equal to 38.0, and S is equal to 11.9. For choice C, we have X bar is equal to 33.5, and S is equal to 11.9. And for choice D, we have X bar is equal to 38.0, and S is equal to 13.2. Now, the first thing we want to do is to calculate the midpoint for each of our class intervals. And we call that to calculate the midpoint, we just add the upper and lower bounds together and then divide by 2. So, we're going to create a 3rd column, and we're going to label it as midpoint for our table. And we'll calculate the midpoint for each of our classes. So for the class from 10 to 19, we get a midpoint of 14.5. For the class interval from 20 to 29, we get a midpoint of 24.5. For 30 to 39, we have a midpoint of 34.5. For 40 to 49, we have a midpoint of 44.5. And then finally for the interval of 50 to 59, we have a midpoint of 54.5. So now we're gonna need to use these midpoints to calculate our sample mean. And so, recall that our sample mean X bar is going to be equal to the sum of the product of our frequency F multiplied by our midpoint X. And then that is going to get divided by the sum of our frequencies. And so that means that X bar is going to be equal to, and the numerator will have our frequency multiplied by our midpoints for each of our class intervals. So, we'll have 4 multiplied by 14.5. Plus 7 multiplied by 24.5. Plus 10 multiplied by 34.5. Plus 6 multiplied by 44.5. And then finally, plus 3 multiplied by 54.5. And that whole quantity is divided by the quantity of 4 + 7 + 10 + 6 + 3. So that means that X bar is going to be equal to. 105 Divided by 30. Which is going to be equal to 33.5. And so that is the first half of our answer that we were looking for. Next, we need to calculate our standard deviation S. And so recall that our standard deviation S is going to be equal to the square root of the quantity of the sum. Of F multiplied by the quantity of X minus X bar in quantity squared divided by the quantity of the sum of F minus 1. And so that means that our standard deviation is going to be equal to the quantity, and here we sum our frequency multiplied by the quantity of X, which is going to be our midpoint, minus X bar which we just calculated, and we're going to square that quantity for each of our class intervals. So in the numerator, we're going to have 4 multiplied by the quantity of 14.5 minus 33.5 in quantity squared. Plus 7 multiplied by the quantity of 24.5 minus 33.5 in quantity squared. Plus 10 multiplied by the quantity of 34.5 minus 33.5. In quantity squared. Plus 6 multiplied by the quantity of 44.5 minus 33.5 in quantity squared. Plus 3, multiplied by the quantity of 54.5. Minus 33.5 and that quantity is going to be squared. And all of that is going to be divided by. The quantity of the sum of F which we found earlier to be 30. 1 in quantity, and all of that is raised to the 1 half power. So here we just rewritten the square root as raised in that quantity to the 1/2 power. And so, evaluating this expression, this is going to be equal to. The quantity Of 4070. Divided by 29 in quantity rates to the 1/2 hour. And when we evaluate this expression, this is going to be approximately equal to. 11.9, and here we just rounded to one decimal place. So that's gonna be the answer for the second half of our problem. So now we've calculated the mean and sample standard deviation. We can compare the values that we calculated to our answer choices, and we see that the correct answer choice corresponds to answer choice C. So I hope that this explanation has been useful, and I'll see you in the next video.

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