Introduction
Market and operations research as a means to gain insights into the various crucial aspects of business operations has gained importance in recent times aiding and supporting managerial decision making. Statistics as a tool of market and operations research and descriptive statistics in particular are playing an increasingly important role in managerial decision making. The present paper is based on a market survey of 150 respondents who were surveyed telephonically and were asked to provide information about themselves, their shopping habits and their view of how the three shopping areas fit into their shopping preferences. The responses were measured against a set of 30 variables grouped into 6 sets (A through F) based on the type of variable. The criteria for grouping the variables into sets included frequency of visit, purchase amount being spent, general attitude, best fit with shopping preferences, relative importance of shopping preferences, and personal information of the respondents .
The paper is specifically concerned with the variables 7 through 9 which measure the general attitude of the respondents towards each mall/shopping area namely, Springdale Mall, Downtown and the West Mall. Additionally, the paper concerns itself with the study of the variable 28, which records the marital status of the respondents. By estimating the point and interval measures of the population through the given sample responses, the paper demonstrates how the business managers can assess their decisions regarding the people’s preferences towards the different malls and help them plan their marketing and operational strategies accordingly. The analysis of the proportion and interval estimates as well as proportion confidence intervals at given level of confidence can help them take decisions with greater accuracy.
Analysis & Discussion
As also indicated above, the Item C in the description of the data collection instrument lists variables 7, 8 and 9, which represent the respondent’s general attitude toward each of the three shopping areas. Each of these variables has numerically equal distances between the possible responses and for purposes of analysis they may be considered to be of the interval scale of measurement. The point estimates for the three variables based were determined to be the population mean estimators in the form of mean sample responses namely µ7, µ8 and µ9, determine the point estimates for the respective variables as mentioned below.
Point estimate for Variable 7
As shown in the table above, the descriptive statistics for the variable 7 or SPRILIKE yield the point estimate as follows:
The point estimate for the population is the sample mean. The same is denoted by X and is calculated as follows
X= n1X
X= 3.83
Interval Estimation for Variable 7 (SPRILIKE)
The sample variance (s2) = [1/(n-1)]*i=1nXi-X
=1.33
The Sample Standard deviation (S) = S^2=1.15
Assuming a normal distribution, the value of population standard deviation can be estimated as follows:
At 95% confidence level, there will be 4 standard deviations
Max=5, Min=1, the range being 4,
The standard deviation for the distribution(x) =Range/4=4/4=1
The Confidence interval = X± sampling error
Confidence interval =X± zx/n
Where, z is the coefficient of standard error at a given level of confidence, x is the population standard deviation and ‘n’ is the sample size.
Here z=1.96 for 95% confidence level,
n= 150 (given)
Therefore,
Confidence interval=3.83±1.96*1/150=3.83±0.16
Since in this case the actual population standard deviation is not known and it has been estimated by non statistical means, the sample standard deviation ‘s’ can be used instead of x.
Thus alternatively,
=> Confidence Interval= X ±zsn=3.83±1.96*1.15/150=3.83±0.18
Thus the confidence interval for µ7 at 95% confidence level can be expressed as
3.83≤ µ≤4.01
What this in effect means is that the chance of the population mean as determined by point estimation, lying between the values 3.83 and 4.01 is 95%.
Point Estimation for variable 8 (DOWNLIKE)
The point estimate for the population is the sample mean. The same is denoted by X and is calculated as follows
X= n1X
X= 3.39
Interval Estimate for µ8
Similarly, the confidence interval for µ8 (DOWNLIKE) is calculated as follows
Sample mean:
X= n1X
X= 3.39
The sample variance (s2) =[ 1/(n-1)]*i=1nXi-X
=1.23
The Sample Standard deviation (S) = S^2=1.11
Confidence interval =X± zx/n
z (the ratio of sample standard error to the population sampling error) for 95% confidence level = 1.96.
In absence of a real value of x, we use S=1.11
Thus, Confidence interval for µ8=3.39 ±1.96*1.11/150 =3.39 ±0.18
Thus the confidence interval for µ8 at 95% confidence level can be expressed as
3.21≤ µ ≤ 3.57
Point Estimate for Variable 9 (WESTLIKE)
Sample mean (Point estimate):
X= n1X
X= 3.09
Interval Estimate for µ9
The confidence interval for µ9 (WESTLIKE) is calculated as follows.
The sample variance (s2) =[ 1/(n-1)]*i=1nXi-X
=1.40
The Sample Standard deviation (S) = S^2=1.18
Confidence interval =X± zx/n
z (the ratio of sample standard error to the population sampling error) for 95% confidence level = 1.96.
In absence of a real value of x, we use S=1.18
Thus, Confidence interval for µ9=3.09 ±1.96*1.18/150 =3.09 ±0.19
Thus the confidence interval for µ9 at 95% confidence level can be expressed as:
≤ µ ≤3.28
Moving ahead, the breakdown of responses for variable 28 (marital status of respondent) was studied, the point estimate was determined and then the 95 percent confidence interval for π28=the population proportion in the “single” or other category was constructed as shown below.
Sample size=z2*(0.25)/(sampling error)2
Sampling error=z2*0.25/n=(1.96)2*0.25/150 =0.08
Also Mean proportion of married persons is given by,
P= No. of Married persons/n=67/150=0.45
Thus the confidence interval for π28= 0.45±0.08
The confidence interval for population proportion p can expressed as
0.37 ≤ p≤0.53
With 95% confidence we can say that from 37% to 53% of the respondents are married.
Further, an assumption was made, that the managers have requested estimates of the mean attitudes towards each mall with a margin of error of 0.05 for each mall. If the managers want to have 95% confidence that the sample mean will fall within this margin of error of 0.05 for each mall, The sample sizes for the respective variables 7, 8 and 9 based on above sampling error, while keeping the point estimates same were calculated as shown below .
Sample size n=z22/ (Sampling Error)2
Assumed Margin of Error for each mall=0.05
Desired Confidence level=95%
For µ7, in absence of population variance we take the value of sample variance=1.33
z=1.96
Sampling error=0.05
Therefore
n= (1.96)2*1.33/0.0025=2044
For µ8, the sample variance as a substitute to population variance = 1.23
Therefore
n= (1.96)2*1.23/0.0025=1890
For µ9, the sample variance as a substitute to population variance=1.40
thus,
n= (1.96)2*1.40/0.0025=2151
The above results show that as the allowed margin of error is reduced, the sample size has to increase to be consistent with the restriction imposed on the results.
Conclusion
As shown above, point estimate for variable 7, 8 and 9 are 3.83, 3.39 and 3.57 respectively. Combined with the confidence interval estimates for the three variables at 95% confidence level (with standard errors of 0.18, 0.18 and 0.19 respectively), it can be concluded that there is more positive attitude towards the Springdale mall and the West mall as compared to the downtown area. This shows greater affinity to organized retail than traditional retail.
Further, with 95% confidence we can say that from 37% to 53% of the respondents are married. This is a substantial proportion and augurs well for the organized retail business.
The results for the sample sizes for variables 7, 8 and 9 upon reducing the allowed error show that as the allowed margin of error is reduced, the sample size has to increase to be consistent with the restriction imposed on the results. In this case, the sample sizes of 2044, 1890 and 2151 are almost 12 to 14 times the current sample size of 150 and might not be feasible for the business managers to conduct the survey at this scale. Thus a 0.05 error margin seems unrealistic.
References
Duchesne, P. (2003). Estimation of a Proportion with Survey Data. Journal of Statistics Education.
Gardner, M. J., & Altman, D. G. (1986). Confidence intervals rather than p values:estimation rather than hypothesis testing. Statistics in Medicine, 746-749.
Hao, C. C., Tung, Y.-K., & Yang, J.-C. (1995). Evaluation of probability point estimate methods. Applied Mathematical Modelling, 95-105.
