Question

question 9 (2*5 points)
consider the data set
4 10 3 7 1 7 3 3 1 7
find the average (mean):
find the median:
question 10 (5 points)
find the 5 number summary for the data shown
26 39 56 58 66 72 88 89 91 93 100
5 number summary:
question 11 (5 points)
a sample was done, collecting the data below. calculate the standard deviation.
2 4 4 4 5 5 7 9
standard deviation =

Explanation:

Question 9 - Mean

Step1: Calculate sum of data

The data set is \(4, 10, 3, 7, 1, 7, 3, 3, 1, 7\). The sum \(S=\sum_{i = 1}^{n}x_{i}=4 + 10+3+7+1+7+3+3+1+7=46\). There are \(n = 10\) data - points.

Step2: Calculate the mean

The mean \(\bar{x}=\frac{S}{n}=\frac{46}{10}=4.6\).

Question 9 - Median

Step1: Sort the data

Sort the data set: \(1,1,3,3,3,4,7,7,7,10\).

Step2: Find the median

Since \(n = 10\) (an even - numbered data set), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th ordered values. The \(\frac{10}{2}=5\)th value is \(3\) and the \((\frac{10}{2}+1)=6\)th value is \(4\). So the median \(M=\frac{3 + 4}{2}=3.5\).

Question 10

Step1: Identify the minimum

The minimum value of the data set \(26,39,56,58,66,72,88,89,91,93,100\) is \(26\).

Step2: Identify the first quartile (\(Q_1\))

There are \(n = 11\) data - points. The position of \(Q_1\) is \(\frac{n + 1}{4}=3\). So \(Q_1 = 56\).

Step3: Identify the median

The position of the median is \(\frac{n+1}{2}=6\). So the median \(M = 72\).

Step4: Identify the third quartile (\(Q_3\))

The position of \(Q_3\) is \(\frac{3(n + 1)}{4}=9\). So \(Q_3 = 91\).

Step5: Identify the maximum

The maximum value is \(100\). The five - number summary is \(26,56,72,91,100\).

Question 11

Step1: Calculate the mean

The data set is \(2,4,4,4,5,5,7,9\). The sum \(S=2 + 4+4+4+5+5+7+9=40\), and \(n = 8\). The mean \(\bar{x}=\frac{40}{8}=5\).

Step2: Calculate the squared differences

\((2 - 5)^2=9\), \((4 - 5)^2 = 1\), \((4 - 5)^2 = 1\), \((4 - 5)^2 = 1\), \((5 - 5)^2 = 0\), \((5 - 5)^2 = 0\), \((7 - 5)^2 = 4\), \((9 - 5)^2 = 16\).

Step3: Calculate the variance

The variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}=\frac{9+1+1+1+0+0+4+16}{7}=\frac{32}{7}\approx4.57\).

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{\frac{32}{7}}\approx2.14\).

Answer:

Question 9 - Mean: \(4.6\)
Question 9 - Median: \(3.5\)
Question 10: \(26,56,72,91,100\)
Question 11: \(2.14\)