9417-2.doc

Course: Understanding of Mathematics and Statistics (9417)
Semester: Spring, 2021
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ASSIGNMENT No. 2
Q. 1 Differentiate between the following:
(i) Descriptive and inferential statistics
Descriptive Statistics
Use descriptive statistics to summarize and graph the data for a group that you choose. This process allows you
to understand that specific set of observations.
Descriptive statistics describe a sample. That’s pretty straightforward. You simply take a group that you’re
interested in, record data about the group members, and then use summary statistics and graphs to present the
group properties. With descriptive statistics, there is no uncertainty because you are describing only the people
or items that you actually measure. You’re not trying to infer properties about a larger population.
The process involves taking a potentially large number of data points in the sample and reducing them down to
a few meaningful summary values and graphs. This procedure allows us to gain more insights and visualize the
data than simply pouring through row upon row of raw numbers!
Inferential Statistics
Inferential statistics takes data from a sample and makes inferences about the larger population from which the
sample was drawn. Because the goal of inferential statistics is to draw conclusions from a sample and generalize
them to a population, we need to have confidence that our sample accurately reflects the population. This
requirement affects our process. At a broad level, we must do the following:
1. Define the population we are studying.
2. Draw a representative sample from that population.
3. Use analyses that incorporate the sampling error.
We don’t get to pick a convenient group. Instead, random sampling allows us to have confidence that the
sample represents the population. This process is a primary method for obtaining samples that mirrors the
population on average. Random sampling produces statistics, such as the mean, that do not tend to be too high
or too low. Using a random sample, we can generalize from the sample to the broader population.
Unfortunately, gathering a truly random sample can be a complicated process.
(ii) Qualitative and Quantitative variable
1. Quantitative Variables: Sometimes referred to as “numeric” variables, these are variables that represent a
measurable quantity. Examples include:
 Number of students in a class
 Number of square feet in a house
 Population size of a city
 Age of an individual
 Height of an individual

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2. Qualitative Variables: Sometimes referred to as “categorical” variables, these are variables that take on
names or labels and can fit into categories. Examples include:
 Eye color (e.g. “blue”, “green”, “brown”)
 Gender (e.g. “male”, “female”)
 Breed of dog (e.g. “lab”, “bulldog”, “poodle”)
 Level of education (e.g. “high school”, “Associate’s degree”, “Bachelor’s degree”)
 Marital status (e.g. “married”, “single”, “divorced”)
(iii) Statistic and Parameter.
Parameters are numbers that summarize data for an entire population. Statistics are numbers that summarize
data from a sample, i.e. some subset of the entire population.
Problems (1) through (6) below each present a statistical study*. For each study, identify both the parameter and
the statistic in the study.
1) A researcher wants to estimate the average height of women aged 20 years or older. From a simple random
sample of 45 women, the researcher obtains a sample mean height of 63.9 inches.
2) A nutritionist wants to estimate the mean amount of sodium consumed by children under the age of 10. From
a random sample of 75 children under the age of 10, the nutritionist obtains a sample mean of 2993 milligrams
of sodium consumed.
3) Nexium is a drug that can be used to reduce the acid produced by the body and heal damage to the
esophagus. A researcher wants to estimate the proportion of patients taking Nexium that are healed within 8
weeks. A random sample of 224 patients suffering from acid reflux disease is obtained, and 213 of those
patients were healed after 8 weeks.
4) A researcher wants to estimate the average farm size in Kansas. From a simple random sample of 40 farms,
the researcher obtains a sample mean farm size of 731 acres.
5) An energy official wants to estimate the average oil output per well in the United States. From a random
sample of 50 wells throughout the United States, the official obtains a sample mean of 10.7 barrels per day.
6) An education official wants to estimate the proportion of adults aged 18 or older who had read at least one
book during the previous year. A random sample of 1006 adults aged 18 or older is obtained, and 835 of those
adults had read at least one book during the previous year.
7) The International Dairy Foods Association (IDFA) wants to estimate the average amount of calcium male
teenagers consume. From a random sample of 50 male teenagers, the IDFA obtained a sample mean of 1081
milligrams of calcium consumed.
8) A sociologist wants to the proportion of adults with children under the age of 18 that eat dinner together 7
nights a week. A simple random sample of 1122 adults with children under the age of 18 was obtained, and 337
of those adults reported eating dinner together with their families 7 nights a week.

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9) A school administrator wants to estimate the mean score on the verbal portion of the SAT for students whose
first language is not English. From a simple random sample of 20 students whose first language is not English,
the administrator obtains a sample mean SAT verbal score of 458.
Q. 2 Consider the data below. This data represents the number of miles per gallon that 30 selected four-
wheel drive sports utility vehicles obtained in city driving.
12 17 16 14 16 18 16 18 17 16 17 15
15 16 16 15 16 19 10 14 15 11 15 15
19 13 16 18 16 20
(a) Construct a frequency distribution using the 4-stepprocedure.
Frequency Distribution Table
Class Count Percentage
10 - 12 3 10
13 - 15 9 30
16 - 18 15 50
19 - 21 3 10
Total 30 100
(b) Deduce cumulative and percentage frequency distributions and Relative frequency
distribution from the distribution in part (a).
Value Frequency Relative Frequency Cumulative Relative Frequency
10 1 0.0333 0.0333
11 1 0.0333 0.0667
12 1 0.0333 0.1000
13 1 0.0333 0.1333
14 2 0.0667 0.2000
15 6 0.2000 0.4000
16 9 0.3000 0.7000
17 3 0.1000 0.8000
18 3 0.1000 0.9000
19 2 0.0667 0.9667
20 1 0.0333 1.0000

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Q.3 You are working for the Transport manager of a large chain of supermarkets which hires cars for
the use of its staff. Your boss is interested in the weekly distances covered by these cars. Mileages
recorded for a sample of hired vehicles from 'Fleet 1' during a given week yielded the following
data:
138 164 150 132 144 125 149 157 161 150
146 158 140 109 136 148 152 144 145 145
168 126 138 186 163 109 154 165 135 156
146 183 105 108 135 153 140 135 142 128
a) Construct a frequency distribution.
100 - 119 4 10
120 - 139 10 25
140 - 159 19 47.5
160 - 179 5 12.5
180 - 199 2 5
Total 40 100
b) Construct a pie chart
c) Construct steam and leaf plot.

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d) Construct histogram and ogive curve.
Classes Frequencies Cum. Frequencies
100 - 119 4 4
120 - 139 10 14
140 - 159 19 33
160 - 179 5 38

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180 - 199 2 40
Q. 4 (a) From a sample of daily production in the yards of 30 carpet looms.
16.2, 15.8, 15.8, 15.8, 16.3, 15.6, 15.7, 16.0, 16.2, 16.1, 16.8, 16.0, 16.4, 15.2, 15.9, 15.9, 15.9, 16.8,
15.4, 15.7, 15.9, 16.0, 16.3, 16.0, 16.4, 16.6, 15.6, 16.9, 16.3
Calculate Mean, Median and Mode from the above ungrouped data
Rearrange:
15.2,15.4,15.6,15.6,15.7,15.7,15.8,15.8,15.8,15.9,15.9,15.9,15.9,16,16,16,16,16.1,16.2,16.2,16.3,16.3,16.3,16.4,
16.4,16.6,16.8,16.8,16.9
Mean ˉx=∑x / n
= 465.5 / 29
=16.0517
Here, n=29 is odd.
Value of 15th observation is median that is 16.
In the given data, the observation 15.9,16 occurs maximum number of times (4)
So mode Z=15.9,16
(b) Calculate Mean, Median and Mode after grouping data given in Part a
15 - 15.4 2 6.9

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15.5 - 15.9 11 37.9
16 - 16.4 12 41.4
16.5 - 16.9 4 13.8
Total 29 100

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Q. 5 The following are the 405 soybean plant heights collected from a particular plot
Plant height 8- 13-17 18-22 23-27 28-32 33-37 38-42 43-47 48-52 53-57

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(cms) 12
No. of
plants
6 17 25 86 125 77 55 9 4 1
Calculate (i) Arithmetic Mean (ii) Geometric Mean, (iii) Harmonic Mean (iv) Median and (v) Mode and
(vi) Check whether the empirical relation among mean, median and mode holds or not.
Class
(1)
Frequency (f)
(2)
Mid value (x)
(3)
d=x-A/h=x-35/5
A=35,h=5
(4)
f⋅d
(5)=(2)×(4)
cf
(7)
8 - 12 6 10 -5 -30 6
13 - 17 17 15 -4 -68 23
18 - 22 25 20 -3 -75 48
23 - 27 86 25 -2 -172 134
28 - 32 125 30 -1 -125 259
33 - 37 77 35=A 0 0 336
38 - 42 55 40 1 55 391
43 - 47 9 45 2 18 400
48 - 52 4 50 3 12 404
53 - 57 1 55 4 4 405
--- --- --- --- --- ---
n=405 ----- ----- ∑f⋅d=-381 -----

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