Question

find the range, variance, and standard deviation for the given sample data, if possible. if the measures of variation can be obtained for these values, do the results make sense? biologists conducted experiments to determine whether a deficiency of carbon dioxide in the soil affects the phenotypes of peas. listed below are the phenotype codes, where 1 = smooth - yellow, 2 = smooth - green, 3 = wrinkled yellow, and 4 = wrinkled - green. 2 1 3 1 4 3 3 3 2 2 2 2 2 2 2 2 1 2 2 2 can the range of the sample data be obtained for these values? choose the correct answer below and, if necessary, fill in the answer box within your choice. a. the range of the sample data is 3 (type an integer or a decimal. do not round.) b. the range of the sample data cannot be calculated. can the standard deviation of the sample data be obtained for these values? choose the correct answer below and, if necessary, fill in the answer box within your choice. a. the standard deviation of the sample data is (round to one decimal place as needed.) b. the standard deviation of the sample data cannot be calculated.

Explanation:

Step1: Calculate the mean

First, count the number of data points \(n = 21\).
The data set \(x=\{2,1,3,1,4,3,3,3,2,2,2,2,2,2,2,1,2,2,2\}\)
The sum of the data \(\sum x=2 + 1+3 + 1+4+3+3+3+2+2+2+2+2+2+2+1+2+2+2\)
\(\sum x = 42\)
The mean \(\bar{x}=\frac{\sum x}{n}=\frac{42}{21}=2\)

Step2: Calculate the variance

The formula for sample variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\)
\((x_{1}-\bar{x})^{2}=(2 - 2)^{2}=0\)
\((x_{2}-\bar{x})^{2}=(1 - 2)^{2}=1\)
\((x_{3}-\bar{x})^{2}=(3 - 2)^{2}=1\)
\((x_{4}-\bar{x})^{2}=(1 - 2)^{2}=1\)
\((x_{5}-\bar{x})^{2}=(4 - 2)^{2}=4\)
\((x_{6}-\bar{x})^{2}=(3 - 2)^{2}=1\)
\((x_{7}-\bar{x})^{2}=(3 - 2)^{2}=1\)
\((x_{8}-\bar{x})^{2}=(3 - 2)^{2}=1\)
\((x_{9}-\bar{x})^{2}=(2 - 2)^{2}=0\)
\((x_{10}-\bar{x})^{2}=(2 - 2)^{2}=0\)
\((x_{11}-\bar{x})^{2}=(2 - 2)^{2}=0\)
\((x_{12}-\bar{x})^{2}=(2 - 2)^{2}=0\)
\((x_{13}-\bar{x})^{2}=(2 - 2)^{2}=0\)
\((x_{14}-\bar{x})^{2}=(2 - 2)^{2}=0\)
\((x_{15}-\bar{x})^{2}=(2 - 2)^{2}=0\)
\((x_{16}-\bar{x})^{2}=(1 - 2)^{2}=1\)
\((x_{17}-\bar{x})^{2}=(2 - 2)^{2}=0\)
\((x_{18}-\bar{x})^{2}=(2 - 2)^{2}=0\)
\((x_{19}-\bar{x})^{2}=(2 - 2)^{2}=0\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=0 + 1+1+1+4+1+1+1+0+0+0+0+0+0+0+1+0+0+0=11\)
\(s^{2}=\frac{11}{21 - 1}=\frac{11}{20}=0.55\)

Step3: Calculate the standard deviation

The formula for sample standard deviation \(s=\sqrt{s^{2}}\)
\(s=\sqrt{0.55}\approx0.7\)

Answer:

A. The range of the sample data is \(3\).
A. The standard deviation of the sample data is \(0.7\).