# STA 2023 Statistics and the Empirical Rule with Sample Data Problems

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Enpury_B

Mathematics

STA 2023

STA

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Sample Data, Statistics
and the Empirical Rule

Introduction :
Random variable is areal valued function define over a sample space. It says, it is a number
associated with each element in a sample space. Sample data gain from the sample space.
Sample space contains all possible outcomes that occurs in a experiment and we called sample
space a population. When we select few outcomes from the sample space, we called that subset
as sample.

Table 1: Sorted Set of Sample Data

Column1
17.9
21.4
23.6
25.8
26.4
26.9
27.2
27.4
28
28.2

Column2
28.3
28.9
29.1
29.1
29.2
29.3
29.3
30.8
31
31.1

Column3
31.3
31.4
31.6
31.6
32
32.3
32.5
32.8
33.3
33.5

Column4
33.7
33.8
33.9
34
34.7
34.9
34.9
37.6
38.2
38.3

Column5
38.3
38.4
38.6
39
39.5
41.1

Table 2 : Frequency Distribution

Class intervals
15-20
20-25
25-30
30-35
35-40
40-45

Frequency
1
2
14
20
8
1

Graph 1 : Histogram

Frequency
25
20
15
10

5
0
15-20

20-25

25-30

30-35

35-40

Table 3: : Frequency Distribution with Midpoints
Class intervals
15-20
20-25
25-30
30-35
35-40
40-45

Mid point
17.5
22.5
27.5
32.5
37.5
42.5

Frequency
1
2
14
20
8
1

40-45

Table 4 : Summary Statistics

Class
intervals
15-20
20-25
25-30
30-35
35-40
40-45

Mid
point(d)* Frequency(f)
17.5
1
22.5
2
27.5
14
32.5
20
37.5
8
42.5
1
46

Ʃfd = 1440

fd
d-x ̅
17.5
-13.80
45
22.50
385
-3.80
650
1.20
300
6.20
42.5
11.20
1440

Ʃf=46

Ʃ𝑓𝑑
𝑥̅ =
Ʃ𝑓
𝑥̅ =31.304
Ʃ𝑓(𝑑 − 𝑥̅ )2 = 1009.24
Ʃ𝑓(𝑑 − 𝑥̅ )2
𝜎2 =
Ʃ𝑓
𝜎 2 = 21.94

σ = 4.68
46 𝑡𝑒𝑟𝑚

Term of 𝑀𝑒𝑑𝑖𝑎𝑛 = ( 2 )

= 23𝑟𝑑 𝑡𝑒𝑟𝑚

Term of Median = 30-35 interval

𝑛
−𝑐
𝑀𝑒𝑑𝑖𝑎𝑛 = 𝑙 + (2
)ℎ
𝑓

Column1
190.44
154.88
202.16
28.8
307.52
125.44
1009.24

cumulative
frequency
1
3
17
37
45
46

ℓ = lower limit of median class interval
C = cumulative frequency preceding to the median class frequency
f = frequency of the class interval to which median belongs
h = width of the class interval
n = sum of frequencies

Median = 26.50
Modal class = 30-35

Mean(𝑥̅ )
Standard deviation(𝜎 2 )
Variance(σ)
Median
Modal class

31.30
21.94
4.68
26.5
30-35

Table 5 : The Empirical Rule
Empirical rule
%
68
95
99.7

Number
of data
31.28
43.7
45.86

Summary :
A population includes all of the elements from a set of data. A sample consists one or more
observations gain by the population. A frequency distribution is an overview of all distinct
values some variable and the number of time they occur. Frequency distribution tells how

frequencies are distributed over values. When making frequency distribution, determine the
range(difference between the highest and lowest observations in data) and decide the number of
classes to estimate approximate size of the interval and then make the class intervals and also
make frequencies according to them.
Histograms are usually presented with vertically rectangular bars with no gap between the
bars. The area under the graph in a histogram represent the total frequency. The X axis gives the
class boundaries and the frequencies is plotted in the Y axis in a histogram. The mid points of
each of the bar in histogram give a bell shaped arc. Since normal distribution is very popular to
model continuous data, histogram may be helpful to understand the data normality.
When discuss the measure of center that measurements, called measures of central tendency.
They are mean, median and mode. Comparison between this three, mean shows as the best
measurement but it also has some disadvantages and median and mode play better roll in
understanding the location of sample data observations. Measures of variation ,called measures
of dispersion. They are standard deviation, range and inter quartile range. Standard deviation is
the most reliable measure. Range is a very poor measure for dispersion. Random number
generator is sometimes used for model sample data with population ones.
The empirical rule is a statistical rule which states that for a normal distribution, almost all
observed data will fall withing three standard deviation of the mean. In particular the empirical
rule predict that 68% of observations fall within the first standard deviation(μ ± σ), 95% within
the first two standard deviations(μ ± 2σ) and 99.7% within the first three standard deviations
(μ ± 3σ).Skewness and symmetry can be identified using the graphical techniques such as
histograms, which clearly shows the shape of a data distributions.

Sample Data, Statistics
and the Empirical Rule

Introduction :
Random variable is areal valued function define over a sample space. It says, it is a number
associated with each element in a sample space. Sample data gain from the sample space.
Sample space contains all possible outcomes that occurs in a experiment and we called sample
space a population. When we select few outcomes from the sample space, we called that subset
as sample.

Table 1: Sorted Set of Sample Data

Column1
17.9
21.4
23.6
25.8
26.4
26.9
27.2
27.4
28
28.2

Column2
28.3
28.9
29.1
29.1
29.2
29.3
29.3
30.8
31
31.1

Column3
31.3
31.4
31.6
31.6
32
32.3
32.5
32.8
33.3
33.5

Column4
33.7
33.8
33.9
34
34.7
34.9
34.9
37.6
38.2
38.3

Column5
38.3
38.4
38.6
39
39.5
41.1

Table 2 : Frequency Distribution

Class intervals
15-20
20-25
25-30
30-35
35-40
40-45

Frequency
1
2
14
20
8
1

Graph 1 : Histogram

Frequency
25
20
15
10
5
0
15-20

20-25

25-30

30-35

35-40

Table 3: : Frequency Distribution with Midpoints
Class intervals
15-20
20-25
25-30
30-35
3...

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