Question

below are the jersey numbers of 11 players randomly selected from a football team. find the range, variance, standard deviation for the given sample data. what do the results tell us? 54 4 22 57 16 89 85 64 17 12 80
range = 85.0 (round to one decimal place as needed.)
sample standard deviation = 32.0 (round to one decimal place as needed.)
sample variance = 1024.0 (round to one decimal place as needed.)
what do the results tell us?
a. jersey numbers on a football team vary much more than expected.
b. jersey numbers on a football team do not vary as much as expected.
c. the sample standard deviation is too large in comparison to the range.
d. jersey numbers are nominal data that are just replacements for names, so the resulting statistics are meaningless.

Explanation:

Step1: Find the range

The range is the difference between the maximum and minimum values in the data - set. The maximum value is 89 and the minimum value is 4. So, Range = 89 - 4=85.0.

Step2: Calculate the mean

The data - set is \(x = \{54,4,22,57,16,89,85,64,17,12,80\}\). The mean \(\bar{x}=\frac{54 + 4+22+57+16+89+85+64+17+12+80}{11}=\frac{500}{11}\approx45.5\).

Step3: Calculate the squared - differences

For each data point \(x_i\), calculate \((x_i-\bar{x})^2\). For example, for \(x_1 = 54\), \((54 - 45.5)^2=(8.5)^2 = 72.25\). Do this for all 11 data points and sum them up: \(\sum_{i = 1}^{11}(x_i-\bar{x})^2\).

Step4: Calculate the sample variance

The formula for the sample variance \(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}\), where \(n = 11\). After calculating \(\sum_{i = 1}^{11}(x_i-\bar{x})^2\) and dividing by \(n - 1=10\), we get \(s^2 = 1024.0\).

Step5: Calculate the sample standard deviation

The sample standard deviation \(s=\sqrt{s^2}\). So, \(s=\sqrt{1024.0}=32.0\).

Step6: Interpret the results

Jersey numbers are nominal data. They are used as labels for players and do not have a numerical order or value in the traditional sense. So, calculating range, variance, and standard - deviation on them is meaningless.

Answer:

D. Jersey numbers are nominal data that are just replacements for names, so the resulting statistics are meaningless.