Practice Questions And Answers On Statistical Concepts

Question 1

1. a. Frequency, relative frequency, percentage frequency with class limit points and mid points are evaluated and represented in the frequency table below (Black, 2009)

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 Table 1: Frequency Distribution table

$120.00 $170.00  $145.00 8 8 0.16 0.16 16%
$170.00 $220.00  $      195.00 23 15 0.3 0.46 30%
$220.00 $270.00  $      245.00 35 12 0.24 0.7 24%
$270.00 $320.00  $      295.00 39 4 0.08 0.78 8%
$320.00 $370.00  $      345.00 44 5 0.1 0.88 10%
$370.00 $420.00  $      395.00 46 2 0.04 0.92 4%
$420.00 $470.00  $      445.00 48 2 0.04 0.96 4%
$470.00 $520.00  $      495.00 50 2 0.04 1 4%

b. The above frequency table was used to draw the histogram and help of MS Excel has been taken for the purpose of drawing the diagram with percentage frequencies.


                                                                                                                         Figure 1: Histogram of the percent frequency distribution

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c. The nature of the histogram was not normal, rather right skewed, which reflected that the data in the data set of Missy Walters were accumulated with high frequencies in the left tail. Median is used as a measure of central tendency when distribution is not normal in nature and hence it (median) was the best choice of measure of location. Mean can be used for any data set without outliers, but median is considered as a better choice as measure and was the appropriate selection as measure of location for this scenario (Montgomery, Runger & Hubele, 2009).

2.a. The regression equation was


The predictive model between demand and unit sale price was obtained from the above ANOVA and regression table. Due to increase in one unit price, it was noticed that demand abridged by 2.14 units, in the regression model. The adverse relation of demand and unit price was earlier explained by Economic theories.

b. Coefficient of determination was known to be

and was calculated accordingly. The completed ANOVA table was used, where SSE = 3132.66, SST = 8181.48,

It was inferred that unit price was an independent variable and was able to explain 62.0% variance of the dependent variable (demand).

c. The evaluation of the correlation coefficient was obvious and calculated as 

it is to be noted that the negative sign was used, as the correlation coefficient and regression coefficient were considered of same sign .The correlation coefficient defined that there was statistically significant correlation between unit price and demand. The nature of the association was highly negative where unit price was a significant factor in measuring demand of an article.

3. a. Calculations were done to fill up the blank spaces in the ANOVA table using MS Excel as provided underneath.

Table 2: ANOVA Table for three treatments

 Source of variation Sum of squares Degrees of freedom Mean square f p
Between Treatments



 195.29  2.89  0.00
Within Treatments (Error)  158.4 21  7.54    
Total  548.98



The given values of SSE, SSB and SST were used for further calculations where degrees of freedom total was 11. For existence of three treatments m = 3, and for 24 observations n = 24. Therefore the total degree of freedom was n-1 = 23. Therefore the values were, SSE = 158.4, SSB = 390.58, SST = 548.98. Consequently calculated values were, MSB = SSB/ (m-1) =390.58/ 2 =195.29, MSE = SSE/ (n-m) = 158.4/ 21 =7.54, F = MSB / MSE = 195.29/ 7.54 = 25.89. The conforming significance level was less than 0.05 and it was observed that the outcome was significant at  level of significance. The F value was 25.89 which at  was in the critical region. Consequently the null hypothesis considering the three treatments to be equally effective was rejected (Heiberger & Neuwirth, 2009).

4. The ANOVA and Regression tables were completed using MS Excel as follows.

Table 3: ANOVA and Regression Data


  Df SS MS f p
Regression 2 40.7 20.35 80.1181102 0.000593
Residual 4 1.016 0.254    
Total 6 41.716 20.604    

Regression Table

  Coefficients Standard Error t p
Intercept 0.8051 0.0698 11.54 0.00
X1 0.4977 0.4617 1.08 0.33
X2 0.4733 0.0387 12.23 0.00

The calculations used to complete the incomplete tables have been given below:

Total observations was, N = 7, consequently total degree of freedom was, N – 1 = 6. Existence of two independent variables indicated degrees of freedom for regression as 2, for residual DF was 4 (6 – 2 = 4). Currently, SSB = 40.7 and SSE = 1.016, so SST = (40.7 + 1.016) = 41.716. MSB = SSB /2 = 20.35, MSE = SSE/ 4 = 0.254. MST = MSB + MSE. The calculation of F value was done as the ratio between MSB and MSE. Significant value was evaluated using the F-distribution function (Albright, Winston & Zappe, 2010).

Standard error of intercept in regression analysis was found as

values were calculated as the proportion of coefficients and standard errors. The p-values were found using the t-distribution function.

a. Using the completed regression table, the estimated regression model was evaluated as

b. The null hypothesis: H0: There was no significant relation between mobiles sold and the independent factors, price and advertising spots.

c. The alternate hypothesis: HA: There was significant relation between mobiles sold and the independent factors, price and advertising spots.

Using the Regression table t-values for and variables were 1.08 (with p-value = 0.33) and 12.23 (with p-value < 0.05), it was apparent that variable X1 did not have significant correlation with number of phones sold per day (Y). Therefore, the estimated regression model was not statistically significant (Draper & Smith, 2014).

d. For the validity of association of the two regression coefficients, hypotheses were tested.

For the coefficient β1, the null hypothesis was: H0: β1 = 0 and alternate hypothesis was: HA: β1≠0.

The result was estimated using t-test where the statistic was evaluated as 

(with p-value =0.33). Therefore the null hypothesis could not be rejected at  level of significance. The first coefficient of price was not significantly different from zero (Cameron & Trivedi, 2013).

For the coefficient β2, let the null hypothesis was: H0: β2 = 0 and alternate hypothesis: HA: β2≠0.

The result was estimated using t-test where statistic was evaluated as

(with p-value < 0.05). Therefore the null hypothesis was rejected at  level of significance. The coefficient was significantly different from zero.

e. The slope or intercept of the regression model was 0.8051 and the statement was that the coefficient had positive sign. Therefore it was inferred that mobiles sold per day had a significant sales figure when the independent influences were kept at zero.