Question

Please the formula in the second pic

Given the following data: 3,2,3, 4,5,3, 4, 6, 4, 10, 7, 11 (1) (2 points) Calculate the sample mean by hand using its definition. Round your intermediate results to two decimal places and your final answer to one decimal place. Same for the remaining questions. (2) (2 points) Calculate the median of the data. (3) (3 points) Calculate the sample variance by hand using its definition. (4) (3 points) Calculate the sample variance by skipping the sample mean and using the S formula that we covered in Lecture 2 (Slide 41). (5) (2 points) Calculate the sample standard deviation. (6) (3 points) Calculate the interquartile mean (i.e., trimmed mean with % 25%).

Answer 1

Answer:

$\sum x =62, \ \sum x^2= 410$

1).

$\bar x =\frac{\sum x}{n}$

$\bar x =\frac{62}{12} =5.17$

2).

Sorted data

data
2
3
3
3
4
4
4
5
6
7
10
11

Median = average of 6^th and seventh values in the sorted data.

Median = (4+4)/2 = 4

3).

Sample variance

data	x-mean	(x-mean)^2
3	-2.2	4.84
2	-3.2	10.24
3	-2.2	4.84
4	-1.2	1.44
5	-0.2	0.04
3	-2.2	4.84
4	-1.2	1.44
6	0.8	0.64
4	-1.2	1.44
10	4.8	23.04
7	1.8	3.24
11	5.8	33.64
	Total	89.68

$Sample \ variance = \sum(x-\bar x)^2 /n-1$

Sample variance = 89.68/11=8.1527

4).

$variance = \frac{\sum x^2- (\sum x)^2 /n)}{n-1}$

$= \frac{410 - ( 62)^2 /12}{12-1}$

= 8.1515

5).

Sample standard deviation = sqrt( 8.1515)

=2.855

6).

Sorted data

data
2
3
3
3
4
4
4
5
6
7
10
11

Interquartile mean = average of middle 6 values

= (3+4+4+4+5+6)/6

=4.3333