Statistics: Frequency Distribution, Regression Analysis, and Hypothesis Testing

This article covers topics such as frequency distribution, relative frequency distribution, percent frequency distribution, regression analysis, and hypothesis testing in statistics.

Statistics
Name:
Institution:
24th May 2018

Question 1:
a. Prepare a frequency distribution, relative frequency distribution, and percent frequency
distribution for the data set using a class width of $50.
Solution
Class Frequency Relative
frequency
Percent
frequency
120-169 8 0.16 16
170-219 15 0.3 30
220-269 12 0.24 24
270-319 4 0.08 8
320-369 5 0.1 10
370-419 2 0.04 4
420-469 2 0.04 4
470-519 2 0.04 4
Total 50 1.00 100%
b. Construct a histogram showing the percent frequency distribution of the furniture-order values
in the sample. Comment on the shape of the distribution.
Solution
The figure below presents the histogram for the furniture order. As can be seen, the distribution
of the data is skewed, particularly it is skewed to the right (that is, it has a much longer tail to
the right).
c. Given the shape of the distribution in part b, what measure of location would be most
appropriate for this data set?
Solution
Since the data is skewed, median would be the best measure of location.

Question 2
a. Determine whether or not demand and unit price are related. Use α = 0.05.
Solution
The hypothesis to be tested is;
The null hypothesis is rejected if the absolute value of computed t-value is greater than the
absolute critical value. In this case, the computed t-value was greater hence we reject the
null hypothesis and by rejecting the null hypothesis we conclude that the beta coefficient for
the unit price is significantly different from zero hence the two variables (demand and unit
price) are significantly related.
b. Compute the coefficient of determination and fully interpret its meaning
Solution
The coefficient of determination is 0.6171; this tells us that 61.71% of the variation in the
dependent variable (y) is explained by the independent variable (x) in the model.
c. Compute the coefficient of correlation and explain the relationship between demand and
unit price.
Solution
Since the beta coefficient for the unit price (X) was negative it therefore means that we take
the negative value of r. hence the coefficient of correlation is -0.7856; this implies that a
strong negative correlation exists between the dependent variable (y) and the independent
variable (x).
Question 3
Using α = .05, test to see if there is a significant difference among the means of the three
populations. The sample sizes for the three treatments are equal.
Source of Variation SS df MS F
Between Treatments 390.58 2
Within Treatments (Error) 158.40 21
Total 548.98 23
The F-critical value is 3.4668 while the computed F value is 25.89. Since the
computed F-value is greater than the critical value thus we reject the null hypothesis

and conclude that there is a significant difference among the means of the three
populations.
Question 4
a. Develop an estimated regression equation relating y to x1 and x2.
Solution
b. At α = 0.05, test to determine if the estimated equation developed in Part a represents a
significant relationship between all the independent variables and the dependent
variable
Solution
We compute the F value
Source of
Variation
df SS MS F
Regression 2 40.70
Residual 4 1.016
The F-critical value is 6.94 while the computed F value is 80.12. Since the
computed F-value is greater than the critical value thus we reject the null
hypothesis and conclude that that the estimated equation developed in Part a
represents a significant relationship between all the independent variables and the
dependent variable.
c. At α = 0.05, test to see if β1 and β2 is significantly different from zero.
Solution
For x1 we have;
The computed t value is less than the critical value, we therefore fail to reject the null
hypothesis and conclude that β1 is not significantly different from zero.
For x2 we have;

The computed t value is greater than the critical value, we therefore reject the null
hypothesis and conclude that β2 is significantly different from zero.
d. Interpret slope coefficient for X2
Solution
The coefficient for x2 is 0.4733; this means that increasing x2 by one unit would
result to an increase in the dependent variable (y) by 0.4733. Similarly, decreasing
x2 by one unit would result to a decrease in the dependent variable (y) by 0.4733.
e. If the company charges $20,000 for each phone and uses 10 advertising spots,
how many mobile phones would you expect them to sell in a day?
Solution
The regression equation is given as follows;
To obtain the number of phones then we need to input the values of x1 and x2.