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 0.7
(round to one decimal place as needed.)
b. the standard deviation of the sample data cannot be calculated.
can the variance 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 variance of the sample data is
(round to one decimal place as needed.)
b. the variance of the sample data cannot be calculated.

Explanation:

Step1: Calculate the range

The range is calculated as \( \text{Range}=\text{Max}-\text{Min} \).
From the data set \(2,1,3,1,4,3,3,3,2,2,2,2,2,2,2,2,1,2,2,2\), the maximum value (\(\text{Max}\)) is \(4\) and the minimum value (\(\text{Min}\)) is \(1\). So, \( \text{Range}=4 - 1=3 \).

Step2: Calculate the variance

The formula for the sample variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\), where \(n\) is the number of data points and \(\bar{x}\) is the sample mean.
First, find the mean \(\bar{x}=\frac{2+1+3+1+4+3+3+3+2+2+2+2+2+2+2+2+1+2+2+2}{20}=\frac{40}{20} = 2\).
Then, calculate \((x_{i}-\bar{x})^{2}\) for each \(x_{i}\):
For \(x = 1\): \((1 - 2)^{2}=1\), there are \(3\) values of \(1\), so contribution is \(3\times1 = 3\).
For \(x = 2\): \((2 - 2)^{2}=0\), there are \(12\) values of \(2\), so contribution is \(12\times0=0\).
For \(x = 3\): \((3 - 2)^{2}=1\), there are \(4\) values of \(3\), so contribution is \(4\times1 = 4\).
For \(x = 4\): \((4 - 2)^{2}=4\), there is \(1\) value of \(4\), so contribution is \(1\times4 = 4\).
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=3 + 0+4 + 4=11\).
\(s^{2}=\frac{11}{20 - 1}=\frac{11}{19}\approx0.6\).

Step3: Calculate the standard deviation

The formula for the sample standard deviation \(s=\sqrt{s^{2}}\).
Since \(s^{2}\approx0.6\), \(s=\sqrt{0.6}\approx0.7\) (rounded to one decimal place).

Answer:

• For the range: A. The range of the sample data is \(3\).
• For the variance: A. The variance of the sample data is \(0.6\).
• For the standard deviation: A. The standard deviation of the sample data is \(0.7\).

However, in the context of phenotype codes (which are nominal - used for categorization rather than numerical value in a mathematical sense), while the range, variance and standard deviation can be calculated mathematically, they do not make sense because the numbers \(1,2,3,4\) are just labels for different phenotypes and do not have a true numerical (arithmetic) meaning.