Answer:
(a) Sample mean = 2.42
Sample variance = 0.29
Sample standard deviation = 0.53
(b) Please see the picture below
Step-by-step explanation:
(a) 1. Calculate the sample mean:
To calculate the sample mean take all the times and divide them between the number of items of the sample:
[tex]x=\frac{1.75+1.91+1.92+2.35+2.53+2.62+3.09+3.15}{8}[/tex]
[tex]x=2.42[/tex]
2. Calculate the sample variance:
To calculate the sample variance lets to name the items as the following:
[tex]x_{1}=1.75[/tex]
[tex]x_{2}=1.91[/tex]
[tex]x_{3}=1.92[/tex]
[tex]x_{4}=2.35[/tex]
[tex]x_{5}=2.53[/tex]
[tex]x_{6}=2.62[/tex]
[tex]x_{7}=3.09[/tex]
[tex]x_{8}=3.15[/tex]
So, the formula to calculate the sample varianza is:
[tex]s^{2}=\frac{(x_{1}-x)^{2}+(x_{2}-x)^{2}+(x_{3}-x)^{2}+(x_{4}-x)^{2}+(x_{5}-x)^{2}+(x_{6}-x)^{2}+(x_{7}-x)^{2}+(x_{8}-x)^{2}}{n-1}[/tex]
where n is the number of items of the sample and x is the sample mean.
Replacing values:
[tex]s^{2}=\frac{(-0.67)^{2}+(-0.51)^{2}+(-0.5)^{2}+(-0.07)^{2}+(0.11)^{2}+(0.2)^{2}+(0.67)^{2}+(0.73)^{2}}{7}[/tex]
[tex]s^{2}=\frac{(0.4489+0.2601+0.25+0.0049+0.0121+0.04+0.4489+0.5329}{7}[/tex]
[tex]s^{2}=\frac{(0.4489+0.2601+0.25+0.0049+0.0121+0.04+0.4489+0.5329}{7}[/tex]
[tex]s^{2}=0.29[/tex]
3. Calculate the sample standard deviation:
The standard deviation is the square root of the variance, so:
[tex]d=\sqrt{0.29}[/tex]
[tex]d=0.53[/tex]
(b) (1) To construct a box plot of the data, first sort the data from the smallest to the largest:
1.75 1.91 1.92 2.35 2.53 2.62 3.09 3.15
(2) Find the median of the data.
As n is an odd numer the median will be the mean of the two data of the center:
[tex]median=\frac{2.35+2.53}{2}[/tex]
[tex]median=2.44[/tex]
(3) Find the first quartile:
[tex]firstquartile=\frac{1.91+1.92}{2}[/tex]
[tex]firstquartile=1.915[/tex]
(4) Find the third quartile:
[tex]thirdquartile=\frac{2.62+3.09}{2}[/tex]
[tex]thirdquartile=2.855[/tex]
(5) Draw the median, first and third quartile and make a box. Then draw the smallest and largest values of the data and draw a line to conect the box. (Please see the picture below)
