Conducting Statistical Analysis: Probabilities

Introduction

This paper is devoted to using descriptive statistical analysis to calculate probabilities in the context of consumer behavior. In particular, the initial data for the work is the distribution of respondents’ responses about specific characteristics and attributes of shopping areas in Springdale. The fundamental essence of such a study, which is already the second part of the whole project, is focused on an attempt to study in-depth patterns of consumer behavior to determine patterns that would allow those interested in sales to increase profits (Hoerl & Snee, 2020). The present part of the project is devoted to calculating conditional probabilities and frequency distributions that allow a descriptive look at patterns of such behavior and identify potential patterns.

Tables of Frequency Distributions and Probabilities

Amount of Money Spent

The primary point for calculating probabilities is compiling a frequency distribution table for each of the variables of interest. Within this section, the variables of interest are SPRSPEND (Springdale Mall), DOWSPEND (Downtown), and WESSPEND (West Mall), that is, the amount of money spent for three shopping centers. Table 1 below shows the frequency distributions depending on the amount of money for each of the three variables. The count of quantities acts as a check: each of the variables has 150 values, and the total number for the three distributions is 450.

Table 1. Frequency distributions for three variables

SPRSPEND DOWSPEND WESSPEND TOTAL
$200 or more 5 13 5 23
$150–under $200 12 6 12 30
$100–under $150 3 11 6 20
$ 50–under $100 20 6 11 37
$ 25–under $50 29 16 17 62
$ 15–under $25 38 32 29 99
less than $15 43 66 70 179
TOTAL 150 150 150 450

The probability that respondents spent at least $15 at Springdale Mall corresponds to the second column of Table 1, all values except for the line “less than $15”. This means that the total number of respondents who spent at least $15 at Springdale Mall equals 107 people. Since the total number of respondents for this shopping center was 150, the probability that individuals will spend at least $15 during their trip is 0.71333. According to similar calculations, the same probabilities for Downtown and West Mall were 0.56000 and 0.53333, respectively. In other words, the probability of spending more than $15 was higher for Springdale Mall, Downtown, and West Mall. Looking at the results differently, West Mall was where respondents were more likely to make cheaper purchases.

High Quality of Goods

As part of this part of the analysis, it was proposed to estimate the probability that a randomly selected respondent from the sample believes that there are goods of the highest quality in a particular shopping center (the BSTQUALI variable). Technically, the calculations come down to the same thing: it is necessary to build a distribution table and, according to the conditions of the task, determine the probabilities (SC, 2021). Table 2 shows the frequency distributions for each of the three shopping centers. The total number of responses is again used as a check so that the total value of the records for BSTQUALI is 150.

Table 2. Frequency distribution for BSTQUALI

Springdale Mall Downtown West Mall No Opinion TOTAL
79 38 11 22 150

As can be seen from the table above, the highest frequency of preferences is characteristic of Springdale Mall: this means that customers of this shopping center were more often sure that high-quality goods were sold in this place. In terms of probabilities, for Springdale Mall, it was equal to 0.52667 (=79/150), for Downtown, it was equal to 0.25333 (=38/150), and 0.07333 (=11/150) for West Mall. In other words, respondents were the least likely to believe that high-quality goods could be sold in the West Mall.

The Contingency Table

In addition to classical probabilities, the statistical analysis also offers conditional probabilities, which are based on the comparison of two or more variables. Within the framework of this section, it is proposed to focus on two variables, namely SPRSPEND (the amount of money spent) and RESPGEND (gender). To compile a frequency distribution table, each of the two variables is divided into eigenvalues, after which the frequencies were calculated based on the combination of two conditions. Table 3 shows the frequency distribution table for each shopping center, depending on who the respondent was (male or female) and what amount they spent in shopping centers. An interesting observation is that there are slightly more women compared to men.

Table 3. Conditional frequency distributions

Springdale Mall Downtown West Mall
Male Female TOTAL Male Female TOTAL Male Female TOTAL
$200 or more 4 1 5 6 7 13 3 2 5
$150–under $200 5 7 12 3 3 6 5 7 12
$100–under $150 2 1 3 7 4 11 3 3 6
$ 50–under $100 9 11 20 4 2 6 6 5 11
$ 25–under $50 8 21 29 7 9 16 5 12 17
$ 15–under $25 14 24 38 14 18 32 8 21 29
less than $15 22 21 43 23 43 66 34 36 70
TOTAL 64 86 150 64 86 150 64 86 150

To calculate the probabilities in this case, one should use the conditional probabilities tool (Barone, 2022). For example, in the first case, it is indicated that the respondent is a woman, and she will spend at least $15 while visiting Springdale Mall. To calculate this probability, the first condition (female) is a generative one — there are 86 out of 150 of them, or 0.57333. The second condition is that she spends at least $15, so all values except “less than $15” should be summed up: the total number of respondents who fall under these criteria is 65 out of 150, or 0.43333. Then the conditional probability that a randomly selected respondent will be a woman and spend at least $15 at Springdale Mall is 0.75581 (=0.43333/0.57333). For a man, the same probability will be 0.65625, indicating that women are more likely to shop above $15 at Springdale Mall. According to similar calculations, the probabilities for Downtown are 0.500,000 for women and 0.640625 for men, that is, in Downtown, men are more likely to make purchases for more than $ 15. Finally, for West Mall, the probability for women was 0.58140, and for men, it was 0.46875, indicating a higher probability of a woman making purchases over $ 15. Thus, the probability of making such purchases by a woman decreases in the row from West Mall to Downtown and Springdale Mall and for men from Springdale Mall to Downtown and West Mall.

Conclusion

In this paper, a descriptive statistical analysis was conducted with calculations of frequency distributions, the purpose of which was to calculate probabilities. The practical meaning of the results has value for developing strategies to increase shopping centers’ sales. So, it was found that Springdale Mall is the leader in the context of the probability of making purchases more expensive than fifteen dollars compared to Downtown and West Mall. In addition, according to consumers’ perception, it is in the Springdale Mall that goods of the highest quality can be sold, whereas the quality of goods in the West Mall was perceived as the worst. Finally, from the point of view of conditional probabilities, it was found that women are more likely to make purchases for more than $ 15 in West Mall and men in Springdale Mall.

References

Barone, A. (2022). Conditional probability: Formula and real-life examples. Investopedia. Web.

Hoerl, R. W., & Snee, R. D. (2020). Statistical thinking: Improving business performance. John Wiley & Sons.

SC. (2021). Frequency distribution. Statistics Canada. Web.

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