Scaling Data Sample annual salaries (in thous...

Question

Scaling Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.

42 36 48 51 39 39 42

36 48 33 39 42 45 50

b. Each employee in the sample receives a 5% raise. Find the sample mean and the sample standard deviation for the revised data set.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says the following are the weights in kilograms of a group of packages, and we have the values of 22, 25, 1921, 24, 23, 2022, 26, and 21. If each package's weight is increased by 10%, what are the new sample mean and sample standard deviation? So the first thing we need to do is to calculate our new package weights. And so, since we're increasing each package's weight by 10%, we'll just multiply each of our data points by 1.1. In doing so, we'll get the values of 24.2. 27.5. 20.9 23.1. 26.4. 25.3. 22 24.2. 28.6. And finally, 23.1. So these are the new data points that we're going to use to calculate our sample mean and sample standard deviation. And so recall your formula for the sample mean, which is given by X bar, and that that is going to be equal to the sum of X of I divided by N, where X of I is just an individual data point, and N is the total number of data points that we have, which in this case, we have 10. We'll also need to recall our formula for the sample standard deviation S and then recall that that is going to be equal to the square root of the quantity of the sum. Of the quantity of X of I minus X bar in quantity squared, divided by the quantity of N minus 1 in quantity. So, we need to calculate these two quantities, and this is actually best done using a table to keep track of all of the information. So, on the screen here, we have a table where in the first column, it's labeled I. So here we just numbered our data points, 123456789, and 10. In the second column we have X of I, and this is given in kilograms, and here we just have the new values that we calculated. And so we have 24.2, 27.5, 20.9, 23.1, 26.4, 25.3, 22.0, 24.2, 28.6, and 23.1. Now, in order to calculate our mean, we'll need to sum up these values. And so when we sum up our values Xabi, we get a sum of 245.3. So now we have enough information to calculate our mean. That means that X bar is going to be equal to the sum of XI, which we just calculated as 245.3, and then that's going to be divided by the number of data points that we have, which is 10. And so when we evaluate this, we get that our sample mean is going to be equal to. 24.53, and this has units of kilograms, since Xabi was given in kilograms, that means our mean is also going to have their say in units. So that is going to be our answer for our sample meme. Next, we'll need to calculate our sample standard deviation. And for this, we'll need to also calculate the quantity of X by minus X bar, since that shows up in our formula for the sample standard deviation. So we'll do this in our table. So in the next column we have X by minus X bar, and we'll calculate this quantity for each of our data points. And so we'll get the values of minus 0.33, 2.97, minus 3.63, minus 1.43, 1.87, 0.77, minus 2.53, minus 0.33, 4.07, and minus 1.43. Now, in the next column, we'll need to calculate the quantity of X of I minus X bar in quantity squared, since that's the quantity that we ultimately want to sum. So, we're basically squaring the previous column, and in doing so, we'll get the values of 0.1089, 8.8209, 13.1769. 2.0449, 3.4969, 0.5929, 6.4009. 0.1089, 16.5649, and 2.0449. And again, we'll need to sum up all of those values, and when we calculate that sum, we get 53.360. So now we have enough information to calculate our sample standard deviation. So we'll have S is equal to the square root of the quantity, and in the numerator here we'll have the sum of the quantity of X by minus X in quantity squared. And so that is the sum that we just calculated, which is the 53.360, and then that's going to be divided by the quantity of 10 minus 1. And so, evaluating this expression, this is going to be approximately equal to 2.43, and this also has units of kilograms. So our sample sample standard deviation is going to be equal to 2.43 kg. So for our final answer, we'll have for our sample mean X bar is equal to 24.53 kg, and our sample standard deviation S is going to be equal to 2.43 kg. So that is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Scaling Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.
42 36 48 51 39 39 42
36 48 33 39 42 45 50
b. Each employee in the sample receives a 5% raise. Find the sample mean and the sample standard deviation for the revised data set.

Key Concepts

Sample Mean

Sample Standard Deviation

Percentage Increase

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