Estimating the Sample Mean and Standard Deviation for ...

Question

Estimating the Sample Mean and Standard Deviation for Grouped Data In Exercises 41–44, make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set.

Weekly Study Hours The distribution of the number of hours that a random sample of college students study per week is shown in the pie chart. Use 32 as the midpoint for “30+ hours.”

Pie chart showing the distribution of weekly study hours among college students, with segments labeled by hour ranges and frequencies.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says a color surveys students about the number of books they read for leisure in a semester. The results are grouped as follows, and we're given a table below the text, and in the first column, we have the books read, and we have the groups of 0 to 23 to 56 to 8, and 9+. And in the second column, we have the frequency. And we have the values of 8, 1411, and 4. And the final row of our table, where it is labeled as total, and under the frequency column, we have 37. Estimate the sample mean and sample standard deviation and use 10 as the midpoint for the 9 plus books group. So, in this problem, we need to calculate the sample mean and the sample standard deviation. So we're gonna start off with sample mean, and we call that our sample mean, which is X bar. is equal to the sum. Of F multiplied by X. And that is divided by the sum of F, where F here is going to be our frequency and X is going to be the midpoint of our group. So in order to use this formula, we'll need to calculate the midpoint for each of our groups. And so this is best done by adding another column into our table. So we'll add the column and label at midpoint. And we call it to calculate the midpoint of a group here, we're gonna take the upper bound and add to it the lower bound, and then divide that quantity by 2. And so, we do this for each of our groups, we'll get the midpoint values of 147, and 10, where 10 was the midpoint of the 9 plus group, which we were told to use that as the midpoint in our problem. Now, if we look at our formula for the sample mean, we see that we need to sum the quantity of F multiplied by X. So, we'll calculate F multiplied by X for each of our midpoint and frequency pairs. And so we'll make it another column in our table, label it as F multiplied by X. And when we calculate this quantity for each of our midpoints and frequency pairs, we will get the values of 8, 56, 77, and 40. But we'll need the sum of all of those values for our formula. So, when we add all those up, we will get 181. So we'll put that in the totals row in our table. So now we have all the information that we need to calculate our sample mean. So that means that X bar is going to be equal to when we sum the quantity of F multiplied by X, we have 181, and that is going to be divided by the sum of F, which is the total of our frequencies. And if we look in the column with our frequencies, we see the total is 37, so we'll divide the 181 by 37. And when we evaluate this expression, this is gonna be approximately equal to 4.89. So that is going to be the answer for the first half of our problem. Now, for the second half of our problem, we need to calculate the sample standard deviation, which is labeled as S. So here we call your formula for the sample standard deviation. So this 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. That is divided by. The quantity of the sum of F minus 1 in quantity. So, what we're going to need to do is calculate this quantity of X minus X bar, and squaring that quantity, and then multiply it by our frequency F. So, we're actually going to add two more columns to our table. So, the first column that we're going to add is going to be the quantity. Of X minus X bar and quantity squared. And we'll need to calculate this for each of our mid midpoint and frequencies pairs. In doing so, we will get the values of 15.1468224982. Then we'll have 0.795471146822. Then we'll have 4.44411979547. And then finally we'll have 26.0927684441. But then we'll need to multiply those values by our frequency F. So, in our last column, we'll have F multiplied by the quantity of X minus X bar in quantity squared, and calculating this for each of our midpoint and frequency values, we'll get. 121.174579985, then we'll get 11.1365960555, then we'll have 48.88. 531-775-02. And then finally we'll have 104.371073776. Now, we'll need that sum, so we'll add up the final column there, and we'll get a total of 285.567, 567, 567. So that means our standard deviation S is going to be equal to the square root of the quantity, and here the numerator will have the sum of F multiplied by the quantity of X minus X bar in quantity squared. We found that sum to be 285.567, 567. 567 And that is going to be divided by. The quantity of the sum of F minus 1, which is 37. -1. And when we evaluate this expression, this is going to be approximately equal to. 2.82. That it's going to be the answer for the second half of this problem. So the final answer for this problem is going to be that X bar is equal to 4.89, and S is going to be equal to 2.82. So that is going to be their answers for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Frequency Distribution

Sample Mean

Sample Standard Deviation

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