Question

below are the jersey numbers of 11 players randomly selected from a football team. find the range, variance, and standard deviation for the given sample data. what do the results tell us? 64 81 1 56 89 66 38 49 15 57 8
range = □ (round to one decimal place as needed.)
sample standard deviation = □ (round to one decimal place as needed.)
sample variance = □ (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 are nominal data that are just replacements for names, so the resulting statistics are meaningless.
c. the sample standard deviation is too large in comparison to the range.
d. jersey numbers on a football team do not vary as much as expected.

Explanation:

Step1: Find the range

The range is the difference between the maximum and minimum values. The data set is \(64,81,1,56,89,66,38,49,15,57,8\). The maximum value is \(89\) and the minimum value is \(1\).
\[Range = 89 - 1=88\]

Step2: Calculate the mean

The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 11\) and \(x_{i}\) are the data - points. \(\sum_{i=1}^{11}x_{i}=64 + 81+1+56+89+66+38+49+15+57+8=524\). So, \(\bar{x}=\frac{524}{11}\approx47.6\)

Step3: Calculate the squared - differences

For each data - point \(x_{i}\), calculate \((x_{i}-\bar{x})^{2}\). For example, for \(x_1 = 64\), \((64 - 47.6)^{2}=(16.4)^{2}=268.96\). Do this for all \(11\) data - points and sum them up. \(\sum_{i = 1}^{11}(x_{i}-\bar{x})^{2}=(64 - 47.6)^{2}+(81 - 47.6)^{2}+(1 - 47.6)^{2}+(56 - 47.6)^{2}+(89 - 47.6)^{2}+(66 - 47.6)^{2}+(38 - 47.6)^{2}+(49 - 47.6)^{2}+(15 - 47.6)^{2}+(57 - 47.6)^{2}+(8 - 47.6)^{2}\)
\(=268.96+1115.56+2171.56+70.56+1713.96+338.56+92.16+1.96+1062.76+88.36+1568.16 = 8492.4\)

Step4: Calculate the sample variance

The sample variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\). Here, \(n = 11\), so \(s^{2}=\frac{8492.4}{11 - 1}=\frac{8492.4}{10}=849.2\)

Step5: Calculate the sample standard deviation

The sample standard deviation \(s=\sqrt{s^{2}}\). So, \(s=\sqrt{849.2}\approx29.1\)

The results tell us about the spread of the jersey numbers. Jersey numbers are nominal data (they are just labels for players), and calculating variance and standard - deviation for nominal data is not very meaningful as these statistics are more appropriate for numerical data where the values have a numerical relationship. So the answer to "What do the results tell us?" is that jersey numbers are nominal data that are just replacements for names, so the resulting statistics are meaningless.

Answer:

Range = 88
Sample variance = 849.2
Sample standard deviation = 29.1
What do the results tell us? B. Jersey numbers are nominal data that are just replacements for names, so the resulting statistics are meaningless.