Question

2.122 largest and smallest standard deviation using only the whole numbers 1 through 9 as possible data values, create a dataset with $n = 6$ and $\bar{x}=5$ and with: (a) standard deviation as small as possible. (b) standard deviation as large as possible. using the 95% rule to draw smooth bell - shaped curves in exercises 2.123 to 2.126, sketch a curve showing a distribution that is symmetric and bell - shaped and has approximately the given mean and standard deviation. in each case, draw the curve on a horizontal axis with scale 0 to 10.

Explanation:

Step1: Recall the formula for the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. Given $n = 6$ and $\bar{x}=5$, then $\sum_{i=1}^{6}x_{i}=n\bar{x}=6\times5 = 30$.

Step2: Minimize the standard - deviation

The standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$. To minimize it, we want the data points to be as close to the mean as possible. Since we can only use whole numbers from 1 to 9, a dataset with values that are close to 5 will minimize the sum of squared differences. The dataset $\{4,4,5,5,6,6\}$ has a sum of $4 + 4+5+5+6+6=30$ and the values are close to the mean of 5.

Step3: Maximize the standard - deviation

To maximize the standard deviation, we want the data points to be as far from the mean as possible while still having a sum of 30. The dataset $\{1,1,1,9,9,9\}$ has a sum of $1+1+1 + 9+9+9=30$. The values are as far from the mean of 5 as possible using the numbers 1 - 9.

Answer:

(a) $\{4,4,5,5,6,6\}$
(b) $\{1,1,1,9,9,9\}$