Business, 15.02.2020 05:14 emmaraeschool

11. The effect of transformations of scale on the mean and standard deviation You just completed a small research project for your psychology class concerning the effects of an event that happened two years ago on women’s opinions and actions today. The mean age of participants in your study is 19.2 years with a standard deviation of 2.1 years. As you write up your results, you realize that what matters is the ages of the participants two years ago when the event happened, not their ages now. You decide to subtract 2 from each of your participants’ ages. After you subtract 2 years, the mean age in your sample is17.2 years. The new standard deviation of the ages in your sample is2.1 years. One of the variables you collected was the study participants’ weights in kilograms. The mean weight of participants in your study is 65.3 kilograms with a standard deviation of 7.836 kilograms. Your professor, however, requested that you report this value in pounds. To convert from kilograms to pounds, you multiply by 2.2. After you multiply the weights of your participants by 2.2, the mean weight in your sample is143.7 pounds. The new standard deviation of the weights in your sample is17.2 pounds.

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Part a: When the value of 2 years is subtracted from the individual values of participant's ages, the mean will be reduced by 2 years from 19.2 to 17.2 whereas the standard deviation will remain the same.

Part b: When the individual values of participant's weights are converted to pounds by multiplication with 2.2, the new mean will be a given as 143.7 (2.2*65.3 kg) whereas the standard deviation is 17.2 (2.2*7.83).

Explanation:

Part a:

As a specific value is subtracted from all the values thus the impact is as given as follows. The mean is given as follows

$\mu=\frac{\sum x}{n}\\$

For the specific value k is subtracted from the each value x, this is given as

$\mu'=\frac{\sum (x-k)}{n}\\$

Thus the new mean is given as

$\mu'=\frac{\sum (x-k)}{n}\\\mu'=\frac{\sum (x)}{n}-\frac{\sum (k)}{n}\\\\\mu'=\mu-k\\$

This indicates that the new mean is simply equal to the difference of old mean and the value subtracted.

The standard deviation is given as

$\sigma=\sqrt{\frac{\sum{(x-\mu)}^2}{n-1}$

The the specific value k is subtracted from the each value of x, the new value of standard deviation is given as

$\sigma'=\sqrt{\frac{\sum{((x-k)-\mu')}^2}{n-1}$

The new standard deviation is given as

$\sigma'=\sqrt{\frac{\sum{(x-k-\mu+k)}^2}{n-1}}\\\sigma'=\sqrt{\frac{\sum{(x-\mu)}^2}{n-1}}\\\sigma'=\sigma$

This indicates that the new standard deviation is simply equal to the original standard deviation.

So when the value of 2 years is subtracted from the individual values of participant's ages, the mean will be reduced by 2 years from 19.2 to 17.2 whereas the standard deviation will remain the same.

Part b

As a specific value is multiplied with all the values thus the impact is as given as follows. The mean is given as follows

$\mu=\frac{\sum x}{n}\\$

For the specific value k is multiplied with each value x, this is given as

$\mu'=\frac{\sum (kx)}{n}\\$

Thus the new mean is given as

$\mu'=\frac{\sum (kx)}{n}\\\mu'=k\frac{\sum (x)}{n}\\\mu'=k\mu\\$

This indicates that the new mean is simply equal to the multiple of old mean and the value multiplied with each value.

The standard deviation is given as

$\sigma=\sqrt{\frac{\sum{(x-\mu)}^2}{n-1}$

Then the specific value k is multiplied with each value of x, the new value of standard deviation is given as

$\sigma'=\sqrt{\frac{\sum{(kx-\mu')}^2}{n-1}$

The new standard deviation is given as

$\sigma'=\sqrt{\frac{\sum{(kx-k\mu)}^2}{n-1}}\\\sigma'=\sqrt{\frac{\sum{k^2(x-\mu)}^2}{n-1}}\\\sigma'=\sqrt{\frac{k^2\sum{(x-\mu)}^2}{n-1}}\\\sigma'=k\sqrt{\frac{\sum{(x-\mu)}^2}{n-1}}\\\sigma'=k\sigma$

This indicates that the new standard deviation is simply equal to the multiple of the original standard deviation with k.

So when the individual values of participant's weights are converted to pounds by multiplication with 2.2, the new mean will be a given as 143.7 (2.2*65.3 kg) whereas the standard deviation is 17.2 (2.2*7.83).

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11. The effect of transformations of scale on the mean and standard deviation You just completed a s...

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