Question

use the appropriate formulas and methods to answer the following questions. do not use technology as an aid, unless otherwise stated.
a sample of size $n = 10$ is drawn from a population. the data is shown below.
119.8 127.7 127.7 105.4 127.7
95.3 106.8 97.5 104 84.7
what is the range of this data set?
range =
what is the standard deviation of this data set? (remember, it is a sample.) please report the answer with appropriate rounding, reporting 2 more decimal places than the original data. please, please, please do not calculate the value by hand.
stdev =

Explanation:

Step1: Find the maximum and minimum values

The given data set is: 119.8, 127.7, 127.7, 105.4, 127.7, 95.3, 106.8, 97.5, 104, 84.7. The maximum value $x_{max}=127.7$ and the minimum value $x_{min} = 84.7$.

Step2: Calculate the range

The formula for the range of a data - set is $Range=x_{max}-x_{min}$. So, $Range = 127.7 - 84.7=43$.

Step3: Calculate the sample mean

The sample mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 10$ and $\sum_{i=1}^{10}x_{i}=119.8+127.7+127.7+105.4+127.7+95.3+106.8+97.5+104+84.7 = 1096.6$. Then $\bar{x}=\frac{1096.6}{10}=109.66$.

Step4: Calculate the squared differences

For each data point $x_{i}$, calculate $(x_{i}-\bar{x})^{2}$. For example, for $x_1 = 119.8$, $(x_{1}-\bar{x})^{2}=(119.8 - 109.66)^{2}=(10.14)^{2}=102.8196$. Do this for all 10 data points and sum them up. Let $S=\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}$.

Step5: Calculate the sample variance

The formula for the sample variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$. After calculating $S$ and substituting $n = 10$ into the formula, we get $s^{2}=\frac{S}{9}$.

Step6: Calculate the sample standard deviation

The sample standard deviation $s=\sqrt{s^{2}}$.
Using a calculator for the standard - deviation part:

Answer:

range = 43
stdev $\approx15.48$