Statistics - Frequency Distribution, Linear Regression, ANOVA, Multiple Regression

This article covers topics like frequency distribution, linear regression, ANOVA, and multiple regression in statistics. It includes tables and figures to explain the concepts.

Running Head: STATISTICS
Statistics
Name of the student:
Name of the university:
Course ID:

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Table of Contents
Answer 1..........................................................................................................................................3
Answer 1. a).................................................................................................................................3
Answer 1. b).................................................................................................................................3
Answer 2..........................................................................................................................................4
Answer 2. a).................................................................................................................................4
Answer 2. b).................................................................................................................................4
Answer 2. c)..................................................................................................................................5
Answer 2. d).................................................................................................................................5
Answer 2. e)..................................................................................................................................5
Answer 3..........................................................................................................................................5
Answer 3. a).................................................................................................................................5
Answer 3. b).................................................................................................................................5
Answer 4..........................................................................................................................................6
Answer 4. a).................................................................................................................................7
Answer 4. b).................................................................................................................................7
Answer 4. c)..................................................................................................................................8
Answer 4. d).................................................................................................................................8
Answer 4. e)..................................................................................................................................9

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Table of tables
Table 1: The frequency distribution table of examination scores of 20 selected students..............2
Table 2: First linear regression model.............................................................................................3
Table 3: One-way ANOVA table....................................................................................................4
Table 4: Multiple regression model.................................................................................................6
Table 5: First linear regression model.............................................................................................7
Table of Figures
Figure 1: Histogram of frequency in percentages of examination scores of 20 selected students. .2

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Answer 1.
Answer 1. a)
Class
intervals
Class
boundaries
Frequencie
s
Cumulative
frequencies
Relative
frequencie
s
Cumulative
relative
frequencies
Percent
frequencies
50-59 49.5-59.5 3 3 0.15 0.15 15%
60-69 59.5-69.5 2 5 0.10 0.25 10%
70-79 69.5-79.5 5 10 0.25 0.5 25%
80-89 79.5-89.5 4 14 0.20 0.7 20%
90-99 89.5-99.5 6 20 0.30 1 30%
Total 20 1.00 100%
Table 1: The frequency distribution table of examination scores of 20 selected students
Answer 1. b)
50-59 60-69 70-79 80-89 90-99
0%
5%
10%
15%
20%
25%
30%
35%
15%
10%
25%
20%
30%
Histogram of Examination Scores
Class intervals of score
Frequencies in percenatges
Figure 1: Histogram of frequency in percentages of examination scores of 20 selected
students
The shape of the distribution of Examination score shows-

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 The distribution of examination score is left-skewed.
 For this distribution, mode>median>mode.
 The right tail of the distribution is longer than its left tail.
Answer 2.
Table 2: First linear regression model
Answer 2. a)
The sample size is (40+1) = 41.
Answer 2. b)
In this linear regression model, the p-value of ‘Unit Price’ (X) (0.175156156) is greater
than the level of significance (0.05). Hence, the null hypothesis of insignificant relationship

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between independent and dependent variable cannot be rejected with 95% probability. The
statistical and linear relationship between ‘Unit Price (X)’ and ‘Supply (Y)’ could be established.
Answer 2. c)
The ‘Co-efficient of determination’ of the model is found to be 0.047996. It shows that
4.799% variability of ‘Unit Price (X)’ is explained by ‘Supply (Y)’.
Answer 2. d)
Pearson’s co-efficient of correlation (r) is 0.21908 that refers a weak positive correlation
between ‘Unit price’ and ‘Supply’ as (r is in-between 0.0 to 0.3). Therefore, these two variables
are not significantly linked.
Answer 2. e)
The linear regression model is estimated as, Y = 0.029*X + 54.076
For unit price (X) = $50,000, then, Y = [54.076 + (0.029*50000)] = 1504.076.
The supply is predicted to be 1504.076 units.
Answer 3.
Answer 3. a)

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Table 3: One-way ANOVA table
Answer 3. b)
At the 5% level of significance, the hypotheses are stated as-
Null hypothesis (H0): The mean scores of all the four programs are equal to each other.
Alternative hypothesis (HA): There exists at-least one inequality in the mean scores of the four
programs.
The p-value of the F-statistic (6.140351) is 0.00557 that is less than level of significance
(5%). Hence, the null hypothesis could be with rejected at 95% probability and the alternative
hypothesis cannot be rejected. It could be concluded that the average productivity of workers of
Program C has higher than other three programs. It is recommended that the line programmers of
Program C are more acceptable because of its are more effectiveness.
Answer 4.
The company records the data of weekly sales of products (y) as response factor; unit
price of the competitor’s product (x1) and advertising expenditures (x2) as predictor factor.

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Table 4: Multiple regression model
Answer 4. a)
The Predicted regression equation is-
‘Sales’ (Y) = 3.597615086 + 41.32002219 * ‘Price’ (X1) + 0.013241819 * ‘Advertising’ (X2).
Answer 4. b)
The significance level of the F-test is assumed to be 10%. The p-value of the multiple
regression model is 0.052643614; it is less than 0.1. Hence, the null hypothesis could be rejected

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with 90% probability. Therefore, the multiple regression model is found to be statistically
significant at 10% level of significance.
Answer 4. c)
The p-values of the predictors are respectively- ‘Price’ (0.036289) and ‘Advertising’
(0.969694). The p-value of ‘Advertising’ is greater than 5% and the p-value of ‘Price’ is less
than 5%. The p-values indicates that ‘Price’ is the significant factor and ‘Advertising’ is the
insignificant factor.
Table 5: First linear regression model

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Answer 4. d)
‘Advertising’ being an insignificant factor, it is dropped from the multiple regression
model.
Now, the regression model considers only ‘Sales’ as dependent variable and ‘Price’ as
independent variable. The newly structured linear regression model finds the estimated equation-
‘Sales (Y)’ = 3.58178844 + 41.6030534 * ‘Price (X1)’.
Answer 4. e)
The slope (β) of the linear regression model is 41.6030534. It shows a positive
association between the predictor ‘Price’ and the response ‘Sales’.
For 1 unit growth in ‘Price’, the ‘Sales’ would grow by 41.6 units. Similarly, for 1 unit
decrease in ‘Price’, the ‘Sales’ would reduce by 41.6 units.