Discuss the limitations of your study, what questions remain unanswered, and make suggestions to find the answer for unanswered issues in the project (and for follow-on work); for example, you may consider questions like these: Do the findings make sense? What else would you like to know about the sample data? What other data would you collect if you could? What other analyses would you want to do then.

## Analysis and Methods

The given report has been written by employee working in Planning Department of the Century National Bank. This report is intended to be handed over to Mr. Selig and the management, senior personnel of the Century National Bank. The objective of this report is to highlight the characteristics of the checking account customers considering various additional information in relation to usage of ATM debit card and number of transactions on ATM machine. A random sample of 60 customers has been selected for statistical analysis (using both descriptive and inferential statistical techniques) in relation to a set of questions to be addressed. These questions become imperative in the changing dynamics of banking industry considering the higher incidence of ATM debit card and related transactions on ATM machines.

From the analysis of the given sample data, it is apparent that only about 20% of the customers tend to have more than $ 2,000 in their account. Further, the given sample of checking balance can be approximated as a normal distribution. While it is a possibility that more than half of the customers use debit card but it would not be reasonable to conclude the same with reasonable certainty. Also, there has been no significant deterioration in the checking balance account which still remains at $1,600. While there has been an increase in mean ATM transactions per month but still it has not exceeded 10 but has increased in excess of 9. Further, the checking account balances tend to be dissimilar across branches located in different city. Also, two significant independent variables that have a statistically significant impact on checking account balances are monthly ATM transactions and number of other bank services used.

Based on the given sample data constituting 60 observations, a statistical analysis has been carried out in order to address the various issues raised in the given case study.

- The type of variable along with the level of measurement for the given variables in the data provided is highlighted below:

- The frequency distribution portraying the checking balance is as highlighted in Appendix 1. The relevant histogram based on the above frequency table is also contained in the same Appendix. The balance of the typical customer would be equal to the average value of the checking balance data provided for 60 customers which amounts to $1,499.87 or about $ 1,500.

Probability that customers have more than $2000 balance in their accounts needs to be calculated here.

Sample standard deviation = $596.90 (computed using excel)

The diagrammatic representation of the situation is highlighted in Appendix 2. Further, the computation of the requisite probability as highlighted below has been made on the assumption that the given sample data can be approximated to a normal distribution.

There is 20.05% probability that a randomly selected customer would have a balance more than $2,000 in the account. Considering the above probability, it would be correct to infer that not many customers have more than $2,000 in their checking accounts.

- The mean and median of the checking balances are highlighted below:

## Probability of Account Balances Over $2000

Mean and median of the checking balances for the four branches are shown below:

Clearly, there seems to be differences among the branches. This is most obvious if one compares the Atlanta and Cincinnati branches values which seem to be significantly different from one another and cannot be attributed to chance. However, this is on expected lines considering the checking balance would depend on the underlying economic and demographic factors that tend to vary from city to city.

- The range, standard deviation, first and third quartile of the checking balances is shown below:

The first quartile indicates that 25% of the 60 accounts have checking balances less than or equal to $1123.75. The third quartile indicates that 75% of the 60 accounts have checking balances less than or equal to $1924.25. Besides, range highlights the difference between the lowest value of checking balance and the highest value. The standard deviation tends to capture the deviation of the data values from the mean value. All the above figures tend to highlight the dispersion of the checking balance account based on the given sample data.

- It is reasonable to expect that the given distribution of checking account balances does approximate a normal distribution. This is because there is only a slight negative skew owing to which the median and mean values do not coincide. However, this is quite possible if the outliers especially on the lower side are excluded which would push the mean value closer to the median value of checking balances. Also, excluding the outliers, about 99% of the checking account balance tends to lie between mean +/- two standard deviations and hence it tends to align in accordance with the empirical rule of normal distribution. Besides, there is high concentration of values near the average value. Further, the shape of the resultant curve is also like a bell curve which can become symmetric if the outliers on both sides are excluded. Considering the above aspects, it would be fair to conclude that the checking account balances distribution can be approximated as a normal distribution.
- The computation of 95% confidence interval for the proportion of customers using debit card is highlighted below.

The z value for 95% confidence interval = 1.960

Number of customers who will use debit card for the transaction = 26

Total number of customers = 60

Proportion of customers who will use debit card for the transactions p = 26 /60 = 0.4333

Standard error

Lower limit of 95% confidence interval

Upper limit of 95% confidence interval

Considering the above confidence interval, it can be stated with a confidence of 95% that the proportion of debit card users lies between 30.76% and 55.06%. Thus, based on the computed confidence interval, it is a possibility that more than 50% of the customers use debit card. However, it cannot be said with certainty that the above claim is true since the lower limit of the confidence interval is significantly lower than 0.5.

The computation of the 95% confidence interval for the mean account balance is highlighted below.

Mean checking account balance = $1,499.87

Sample standard deviation = $596.60

Number of observations = 60

Standard error = 596.60/√60 = $77.06

Lower limit of 95% confidence interval

Upper limit of 95% confidence interval

Based on the above confidence interval, it can be concluded with a 95% likelihood that mean checking account balance for all accounts would lie between $1,348.83 and $1650.90.

- The aim is to check the validity of the claim that the mean checking balance is $1600 for the customers who has allowed the money to sit in checking account.

Null hypothesis

Alternative hypothesis

The appropriate test statistic would be T in the given case since the population standard deviation is not known.

## Analysis of Checking Balances per Branch

The value of test statistics can be determined as shown below:

The p value for the inputs 59 degree of freedom and t value of -1.3196 comes out to be 0.09613.

Level of significance = 0.05

It can be seen from the above that p value is higher than the level of significance and thus, insufficient evidence is present for the rejection of null hypothesis. Hence, the conclusion can be drawn that the mean checking balance for a customer is higher than or equal to $1600. Thus, the mean account balance has not declined from the historical level of $ 1,600.

The value of test statistics can be determined as shown below:

Mean number of transaction per month per customer = 10.3

Standard deviation of mean number of transaction = 4.295

The p value for 59 degree of freedom and t statistic of -0.540 comes out to be 0.2956.

Level of significance = 0.05

It can be seen from the above that p value is higher than the level of significance and thus, insufficient evidence present for the rejection of null hypothesis. Hence, the conclusion can be drawn that mean number of transaction per month per customer is not higher than 10.

In order to test whether the mean monthly transaction exclude 9 or not, the 95% confidence interval for the monthly transaction would be computed.

Alternative hypothesis

The t value for 95% confidence interval and 59 degree of freedom (two tailed) =2.000995

Lower limit

Upper limit

The 95% confidence interval

It is apparent based on the above result that number of transaction 9 does not fall in the above calculated confidence interval and thus, the null hypothesis would be rejected. Hence, it can be concluded with 95% confidence that average monthly ATM transactions is higher than 9.

For comparing the mean checking account across four branches, t test cannot be used since it can be used for comparison of means of only two variables. As a result, a single factor ANOVA would be used here which seems appropriate considering the variances are also equal. The requisite hypotheses are highlighted below.

Null Hypothesis: The mean checking account balance across the four branches is the same

Alternative Hypothesis: Atleast one of the branches would have mean checking account balance different from the other three.

The ANOVA output is highlighted in Appendix 3. It is apparent that the relevant p value is 0.015 and assuming a significance level of 5%, it may be concluded that the null hypothesis may be rejected and alternative hypothesis be accepted. Hence, it may be appropriate to conclude that the average checking balance amount is different for atleast one branch.

## Confidence Intervals for Debit Card Usage and Mean Account Balance

For comparison between the specific branches, the relevant output is attached in Appendix 4. Based on the output attached, it is apparent that a significant difference in mean checking balance account tends to exist for Cincinnati and Atlanta. However, the same cannot be concluded about Atlanta and Louisville along with Atlanta and Eric.

Regression model by taking checking account balance as dependent variable and number of ATM transaction as independent variable has been obtained through excel and the relevant output is attached in Appendix 5. The equation of the regression line is given below.

It is apparent that the p value associated with the slope coefficient comes out to be 0.00. This implies that at a significance level of 1%, the slope is significant which implies that number of ATM transactions tend to have a statistically significant impact on checking account balances. Also, the ANOVA output highlights that the p value for the F statistic has come to be zero. This implies that the linear regression model is significant assuming a 1% significance level.

Also, the coefficient of determination or R^{2} has a value of 0.5025 which implies that 50.25% of the changes in the checking account balance can be explained on account of corresponding changes in the number of ATM transactions. Thus, based on the above discussion, it would be appropriate to conclude that additional independent variables need to be included in the given linear regression model so as to improve the R^{2} or predictive power. However, the number of ATM transactions remain sa significant variable for determining the checking account balance which is quite logical considering a proportional relationship between the two.

Regression model by taking checking account balance as dependent variable and number of other bank service as independent variable has been conducted through Excel and the relevant output is attached in Appendix 6. The relevant regression line based on the given output is indicated below.

It is apparent that the p value associated with the slope coefficient comes out to be 0.00. This implies that at a significance level of 1%, the slope is significant which implies that number of ATM transactions tend to have a statistically significant impact on checking account balances. Also, the ANOVA output highlights that the p value for the F statistic has come to be zero. This implies that the linear regression model is significant assuming a 1% significance level. However, considering that the coefficient of determination is lower for this model as compared to the previous model, hence it would be appropriate to conclude that the given model is weaker than the previous model.

Conclusions

The sample provided highlights that only about one-fifth customers have a checking account balance in excess of $ 2,000. Also, the sample data with regards to checking balance can be closely approximated as normal distributed. In relation to the proportion of customers using debit card being greater than 50%, it would be unreasonable to assume the same but it could be a possibility. Additionally, the checking account balances tend to show significant variation across different branches but have not fallen below $ 1,600. Also, the monthly ATM transactions have shown increase from previous levels and have exceeded 9 but failed to exceed 10. Further, regression analysis suggests that two significant factors influencing the checking account balance are other bank services used along with monthly ATM transactions.

A major limitation is that the dataset was small and was skewed. The findings in the analysis of the sample data does indicate a shift towards more usage of ATM machines but the change seems to be happening at a lesser pace than predicted. It would have been worthwhile to conduct the regression analysis for the various branches separately as it is expected that the penetration of debit cards and their usage pattern would not be constant across cities.

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