hw 02 ch 5: problem 3 (12 points)\nfor the following list of data values: 32, 20, 23, 26, 24\na. find the…

Explanation:

Step1: Calculate the average

The formula for the average (mean) $\bar{x}$ of a data - set $x_1,x_2,\cdots,x_n$ is $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here, $n = 5$, $x_1=32,x_2 = 20,x_3=23,x_4=26,x_5=24$. So, $\sum_{i=1}^{5}x_i=32 + 20+23+26+24=125$, and $\bar{x}=\frac{125}{5}=25$.

Step2: Calculate the standard deviation (SD)

The formula for the standard deviation $s$ of a sample is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}$. $(32 - 25)^2=49$, $(20 - 25)^2 = 25$, $(23 - 25)^2=4$, $(26 - 25)^2 = 1$, $(24 - 25)^2=1$. $\sum_{i = 1}^{5}(x_i - 25)^2=49+25 + 4+1+1=80$. Then $s=\sqrt{\frac{80}{4}}=\sqrt{20}\approx4.47$.

Step3: Convert to standard units

The formula for converting a value $x$ to standard units (z - score) is $z=\frac{x-\bar{x}}{s}$. i. For $x = 32$, $z=\frac{32 - 25}{4.47}\approx1.57$. ii. For $x = 20$, $z=\frac{20 - 25}{4.47}\approx - 1.12$. iii. For $x = 23$, $z=\frac{23 - 25}{4.47}\approx - 0.45$. iv. For $x = 26$, $z=\frac{26 - 25}{4.47}\approx0.22$. v. For $x = 24$, $z=\frac{24 - 25}{4.47}\approx - 0.22$.

Step4: Calculate the average of the converted list

The average of a set of z - scores is always 0. Mathematically, if $z_i=\frac{x_i-\bar{x}}{s}$ for $i = 1,\cdots,n$, then $\bar{z}=\frac{\sum_{i = 1}^{n}z_i}{n}=\frac{\sum_{i = 1}^{n}\frac{x_i-\bar{x}}{s}}{n}=\frac{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})}{s}}{n}$. Since $\sum_{i = 1}^{n}(x_i-\bar{x})=0$, $\bar{z}=0$.

Step5: Calculate the SD of the converted list

The standard deviation of a set of z - scores is always 1.

Answer:

a. 25 b. $\sqrt{20}\approx4.47$ c. i. 1.57 ii. - 1.12 iii. - 0.45 iv. 0.22 v. - 0.22 d. 0 e. 1