Banking Statistical Analysis

Question 01

Average account balance by city

Draw a bar chart showing the average Account Balance for customers in each city.

Answer and analysis

Houston has the highest average account balance ($1,879.65), followed by San Lucas ($1,423.46), Dallas ($1,359.36) and Jamestown ($1,281.38). Houston exceeds Jamestown by $598.27 in this sample. The chart describes the sample; Question 8 tests whether the city mean differences are statistically significant.

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Method and hypotheses

Descriptive comparison: mean account balance is plotted for each of the four numeric-coded city groups. No hypothesis test is needed for this question.

Checks and test selection

There are 60 customers: Jamestown 16, Houston 17, Dallas 14 and San Lucas 13. The vertical axis shows the mean, not the total balance.

Question 02

ATM descriptive statistics by city

For each city, find the mean, median, mode, standard deviation, range, maximum, and minimum for number of ATM transactions per month.

Answer and analysis

Houston has the highest average ATM use (11.29 transactions per month). San Lucas is the most variable city (SD = 5.82; range = 18), and its mean (10.08) is above its median (8), reflecting the influence of higher-use customers. The multiple modes show that no city has a single uniquely most frequent transaction count.

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Method and hypotheses

SPSS Frequencies is run separately for each city. Standard deviations are sample standard deviations. All tied modes are included in the summary below.

Checks and test selection

SPSS prints only the smallest mode when several values tie; its footnote is retained in the original table images.

Question 03

Account balance and receiving interest

Test if there is a significant difference between the Account Balance and Receive interest on the account.

Answer and analysis

Yes. Customers receiving interest have a higher mean balance ($1,693.50; SD = $282.51) than customers not receiving interest ($1,429.45; SD = $664.83). Welch’s two-sided t-test gives t(56.424) = -2.153, p = .036. Reject H0 at .05. The difference (no interest minus interest) is -$264.05 (95% CI: -$509.63 to -$18.46). Interest status is associated with balance, but this observational comparison does not show that receiving interest causes a higher balance.

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Method and hypotheses

Independent-samples t-test comparing customers who receive interest with those who do not. H0: the two population mean balances are equal. H1: they differ.

Checks and test selection

Normality is not rejected: no-interest group (n = 44), Kolmogorov–Smirnov p = .200; interest group (n = 16), Shapiro–Wilk p = .077. Levene’s p < .001, so use the “Equal variances not assumed” (Welch) row.

Question 04

Other bank services compared with seven

Test the hypothesis that the average of Number of other bank services used is equal to 7.

Answer and analysis

No. The sample mean is 4.42 services (SD = 1.98), significantly below 7: t(59) = -10.122, p < .001. Reject the mean-equals-7 hypothesis at .05. The estimated shortfall is 2.58 services (95% CI for mean minus 7: -3.09 to -2.07); the mean itself has a 95% CI of 3.91 to 4.93. The course-based Wilcoxon check also rejects its location/median benchmark (standardized statistic = -6.124, p < .001), supporting the direction of the finding. The bank could investigate barriers to using additional services rather than assume every customer needs seven.

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Method and hypotheses

H0: the population mean number of other services is 7. H1: it is not 7. A two-sided one-sample t-test addresses this mean hypothesis, using a large-sample approximation (n = 60). A one-sample Wilcoxon test is also reported to follow the course’s non-normal-data branch.

Checks and test selection

Kolmogorov–Smirnov p = .002 rejects normality under the course rule for n > 30. These are discrete counts. The t-test therefore relies on the sampling-mean approximation, not on normally distributed individual counts. Wilcoxon evaluates a location/median benchmark under its symmetry assumption; it does not test the mean directly.

Question 05

City and debit-card ownership

Test if there is a significance association between the "City where banking is done" and "whether or not the customer has a debit card".

Answer and analysis

No. There is no statistically significant evidence of an association between city and debit-card ownership: χ²(3, N = 60) = 0.224, p = .974. Fail to reject H0 at .05. Cramer’s V = .061 indicates a very small observed association. Debit-card ownership ranges from 38.5% in San Lucas to 47.1% in Houston; those sample differences do not establish different ownership rates in the populations. The bank should not base city-specific debit-card policies on these data alone.

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Method and hypotheses

Pearson chi-square test of independence for a 4 × 2 contingency table. H0: city and debit-card ownership are independent. H1: they are associated.

Checks and test selection

Both variables are nominal. All expected cell counts exceed 5 (minimum = 5.63), so the chi-square approximation is appropriate, assuming independent customers.

City and debit-card ownership: chart

The chart shows counts, not ownership percentages. The city sample sizes differ (16, 17, 14 and 13), so compare ownership percentages as well as bar heights. Jamestown: 7/16 = 43.8%; Houston: 8/17 = 47.1%; Dallas: 6/14 = 42.9%; San Lucas: 5/13 = 38.5%. These figures are consistent with the non-significant chi-square result.

Question 06

Account balance and debit-card ownership

Test the hypothesis that “there is no significance between the Account Balance and whether or not a customer has a debit card.”

Answer and analysis

No. A significant balance difference is not detected: Mann–Whitney U = 415.000, Z = -0.403, SPSS two-sided p = .687. Fail to reject the distributional null hypothesis at .05. Mean balances are $1,435.82 without a debit card and $1,583.62 with one. The Welch mean comparison also fails to reject equality (p = .326); the mean difference (no card minus card) is -$147.79 (95% CI: -$446.23 to $150.65). These results do not prove the groups are identical; debit-card ownership alone is not a reliable basis for predicting balance here.

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Method and hypotheses

The course-based analysis uses a two-sided Mann–Whitney U test because one group fails normality. H0: the balance distributions are the same in the two groups. A Welch t-test is included as a sensitivity check for the narrower question about equality of mean balances.

Checks and test selection

No-card group (n = 34): Kolmogorov–Smirnov p = .044, so normality is rejected. Card group (n = 26): Shapiro–Wilk p = .987. Mann–Whitney does not require normality, but its result cannot automatically be interpreted as a test of means or medians without additional distributional assumptions.

Question 07

Account balance compared with $2,000

Test the hypothesis that the average Account Balance is equal 2000 $.

Answer and analysis

No. The estimated mean balance is $1,499.87 (SD = $596.90), significantly below $2,000: t(59) = -6.490, p < .001. Reject H0 at .05. The estimated mean difference is -$500.13 (95% CI: -$654.33 to -$345.94), and the mean balance has a 95% CI of $1,345.67 to $1,654.06. For planning based on customers like this sample, using $2,000 as the expected average would overstate observed balances.

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Method and hypotheses

Two-sided one-sample t-test. H0: the population mean account balance is $2,000. H1: it differs from $2,000.

Checks and test selection

For n = 60, Kolmogorov–Smirnov p = .176 does not reject normality under the course rule. Shapiro–Wilk p = .063 also does not reject normality. Independence and a representative sample are needed for population inference.

Question 08

Account balance across the four cities

Test if there is a significant difference between the average Account Balance for the different cities.

Answer and analysis

Yes. Welch ANOVA shows that at least one city mean differs: F(3, 28.026) = 6.521, p = .002. Reject H0 at .05. Games–Howell identifies Houston as significantly higher than Jamestown by $598.27 (p = .002; 95% CI: $199.39 to $997.16). No other city pair is significant at .05 after these comparisons. Houston has the highest observed mean, but the result does not establish that banking in Houston causes higher balances; customer composition and other factors could explain the difference.

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Method and hypotheses

One-factor comparison of four independent city groups. H0: all four population mean balances are equal. H1: at least one differs. Use Welch one-way ANOVA and Games–Howell post-hoc comparisons because variances are unequal.

Checks and test selection

Each city has fewer than 30 customers. Shapiro–Wilk p-values are .543 (Jamestown), .839 (Houston), .257 (Dallas) and .936 (San Lucas); normality is not rejected. Levene’s p = .040 rejects equal variances. The ordinary pooled ANOVA row is therefore not the primary test.

City mean comparisons: Games–Howell

The significant pair is Houston–Jamestown. Houston–Dallas has p = .079 and Houston–San Lucas has p = .185, so neither is significant at .05. An omnibus difference does not imply that every city differs from every other city. The repeated reverse-direction rows are the same comparisons with opposite signs; they are not additional independent findings.

Question 09

Excel correlation: balance and ATM transactions

Using Excel, find the correlation coefficient between the Account Balance and Number of ATM transactions per month? Interpret your result.

Answer and analysis

There is a strong positive linear relationship: r = 0.70499 (approximately .705), n = 60, p < .001. Customers with more ATM transactions tend to have higher account balances in this sample. The squared correlation is about .497, so the single-predictor linear relationship accounts for about 49.7% of the observed balance variation. The plotted trendline uses ATM alone; it is not the two-predictor equation in Question 10. Association does not establish that increasing ATM transactions will increase a customer’s balance.

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Method and hypotheses

Pearson correlation calculated in Microsoft Excel: =CORREL(Data!A2:A61,Data!B2:B61). Both variables are quantitative, and the scatterplot is used to assess the direction and approximate linear form.

Checks and test selection

The calculation includes all 60 customer pairs. Correlation describes a linear relationship; independent observations and the usual inferential assumptions are needed for its p-value. The course classifies a correlation above .70 as strong.

Question 10

Excel multiple regression and prediction

Using Excel, estimate: Account Balance = a + b (Number of ATM transactions) + c (Number of other bank services used). (i) Write the linear equation of the model. (ii) Predict Account Balance when ATM transactions per month = 15 and other bank services used = 4.

Answer and analysis

(i) Predicted Account Balance ($) = 247.0307 + 93.1334 × ATM transactions + 68.2241 × other services. (ii) At 15 ATM transactions and 4 services, the predicted balance is $1,916.93, using the unrounded Excel coefficients. Holding the other predictor constant, one additional ATM transaction is associated with $93.13 higher predicted balance, and one additional service with $68.22. The model is significant, F(2, 57) = 34.200, p < .001; R² = .545 and adjusted R² = .530. ATM has p < .001 and services p = .017. The prediction is an estimated conditional mean, not a guaranteed individual balance; residual standard error is about $409.43. These coefficients are associations, not causal effects.

Balance = 247.0307 + 93.1334 × ATM + 68.2241 × services

View method, checks and output images

Method and hypotheses

Ordinary least-squares multiple regression calculated in Microsoft Excel using LINEST with an intercept: =LINEST(Data!A2:A61,Data!B2:C61,TRUE,TRUE). LINEST returns predictor coefficients in reverse column order, so the services and ATM coefficients are assigned to the correct variables. Excel t- and F-distribution formulas calculate inference.

Checks and test selection

The predictors are ATM transactions and other services, not the numeric city or yes/no codes. SPSS model output and residual plots accompany the Excel results. The requested input values (15 ATM transactions, 4 services) are within the observed ranges.

Regression diagnostics

The residual histogram and P-P plot show no pronounced departure from approximate normality. The residual-versus-fitted plot shows no obvious curved pattern or strong funnel, although visual checks cannot prove linearity or constant variance. Both VIFs are 1.054, so there is little evidence of multicollinearity. Standardized residuals range from -2.337 to 2.481 and maximum Cook’s distance is .196: no case exceeds the conventional Cook’s-distance threshold of 1, but influential cases still deserve review. Independence cannot be established from these plots; sampling and collection details remain important.