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?
74 70 94 97 29 38 52 59 77 36 37
range = 68 (round to one decimal place as needed.)
sample standard deviation = 23.9 (round to one decimal place as needed.)
sample variance = \boxed{} (round to one decimal place as needed.)

Explanation:

Step1: Recall the formula for sample variance

The formula for sample variance \( s^2 \) is \( s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1} \), where \( x_i \) are the data points, \( \bar{x} \) is the sample mean, and \( n \) is the number of data points. First, we need to find the sample mean \( \bar{x} \).

The data set is: \( 74,70,94,97,29,38,52,59,77,36,37 \). The number of data points \( n=11 \).

Step2: Calculate the sample mean \( \bar{x} \)

\( \bar{x}=\frac{74 + 70+94 + 97+29+38+52+59+77+36+37}{11} \)
First, sum the data points: \( 74+70 = 144 \); \( 144+94=238 \); \( 238 + 97=335 \); \( 335+29 = 364 \); \( 364+38=402 \); \( 402+52 = 454 \); \( 454+59=513 \); \( 513+77=590 \); \( 590+36 = 626 \); \( 626+37=663 \)
So, \( \bar{x}=\frac{663}{11}\approx60.27 \)

Step3: Calculate \( (x_i-\bar{x})^2 \) for each data point

• For \( x_1 = 74 \): \( (74 - 60.27)^2=(13.73)^2\approx188.51 \)
• For \( x_2 = 70 \): \( (70 - 60.27)^2=(9.73)^2\approx94.67 \)
• For \( x_3 = 94 \): \( (94 - 60.27)^2=(33.73)^2\approx1137.71 \)
• For \( x_4 = 97 \): \( (97 - 60.27)^2=(36.73)^2\approx1349.09 \)
• For \( x_5 = 29 \): \( (29 - 60.27)^2=(- 31.27)^2\approx977.81 \)
• For \( x_6 = 38 \): \( (38 - 60.27)^2=(-22.27)^2\approx495.95 \)
• For \( x_7 = 52 \): \( (52 - 60.27)^2=(-8.27)^2\approx68.39 \)
• For \( x_8 = 59 \): \( (59 - 60.27)^2=(-1.27)^2\approx1.61 \)
• For \( x_9 = 77 \): \( (77 - 60.27)^2=(16.73)^2\approx279.99 \)
• For \( x_{10}=36 \): \( (36 - 60.27)^2=(-24.27)^2\approx589.03 \)
• For \( x_{11}=37 \): \( (37 - 60.27)^2=(-23.27)^2\approx541.59 \)

Step4: Sum up the \( (x_i-\bar{x})^2 \) values

\( 188.51+94.67 = 283.18 \); \( 283.18+1137.71=1420.89 \); \( 1420.89+1349.09 = 2769.98 \); \( 2769.98+977.81=3747.79 \); \( 3747.79+495.95 = 4243.74 \); \( 4243.74+68.39=4312.13 \); \( 4312.13+1.61 = 4313.74 \); \( 4313.74+279.99=4593.73 \); \( 4593.73+589.03=5182.76 \); \( 5182.76+541.59 = 5724.35 \)

Step5: Calculate the sample variance

Using the formula \( s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1} \), with \( \sum(x_i-\bar{x})^2 = 5724.35 \) and \( n = 11 \), so \( n-1=10 \)
\( s^2=\frac{5724.35}{10}=572.435\approx572.4 \) (rounded to one decimal place)

Answer:

\( 572.4 \)