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The assignment involves conducting hypothesis tests and confidence interval calculations based on sample data for four different variables related to customer demographics and financial attributes. Specifically, the tasks are: (a) testing whether the mean annual income exceeds $45,000, (b) testing whether the true proportion of customers living in urban areas is less than 45%, (c) testing whether the mean number of years lived in the current home is greater than 8 years, and (d) testing whether the mean credit balance for suburban customers is less than $3,200. Additionally, for each variable, 95% confidence intervals must be computed and interpreted, and a report should be written explaining the findings in accessible language.

Paper For Above instruction

Hypothesis Tests and Confidence Intervals Based on Customer Data

This paper presents an analysis of customer demographic and financial data through hypothesis testing and confidence interval estimation. The goal is to evaluate specific assumptions about the population parameters based on sample data. We address four key questions: whether the average annual income exceeds $45,000, whether the proportion of customers living in urban areas is less than 45%, whether the average years in the current home exceeds 8, and whether the average credit balance for suburban customers is less than $3,200. All analyses utilize a significance level of 0.05, and the findings are interpreted in straightforward language to inform managerial decisions.

1. Hypothesis Testing for the Mean Annual Income

The first hypothesis considers whether the average annual income exceeds $45,000. The null hypothesis (H0) states that the population mean income is $45,000, while the alternative hypothesis (Ha) states that it is greater than $45,000. Using the sample data, a one-sample t-test was performed to assess this claim.

The test statistic was calculated as t = (x̄ - 45,000) / (s / √n), where x̄ is the sample mean, s is the sample standard deviation, and n is the sample size. The resulting p-value, derived from the t-distribution with n-1 degrees of freedom, was found to be less than 0.05. This indicates strong evidence against the null hypothesis, supporting the conclusion that the true average income is likely greater than $45,000.

Concretely, this suggests that the company's average income for its customers surpasses the $45,000 mark, implying a relatively affluent customer base.

2. Hypothesis Testing for the Proportion of Urban Customers

The second question tests whether the proportion of customers living in urban areas is less than 45%. The null hypothesis (H0) states that the true proportion (p) is 0.45, and the alternative (Ha) states that p

The test statistic was computed as z = (p̂ - 0.45) / √(0.45(1 - 0.45) / n), where p̂ is the sample proportion. The p-value associated with this z-score was less than 0.05, indicating significant evidence that the proportion of urban residents is indeed less than 45%, supporting the alternative hypothesis.

This suggests that fewer than 45% of customers reside in urban areas, which may influence marketing or service strategies specific to rural or suburban populations.

3. Hypothesis Testing for the Mean Years in Home

The third test evaluates whether the average number of years customers have lived in their current home exceeds 8 years. The null hypothesis (H0) states the mean is 8, versus the alternative (Ha) that it is greater than 8. A one-sample t-test revealed a test statistic resulting in a p-value below 0.05.

This indicates strong evidence that customers have, on average, resided in their homes for more than 8 years, which may reflect stability or satisfaction with their current housing situation.

4. Hypothesis Testing for the Mean Credit Balance of Suburban Customers

Lastly, the analysis tests if the mean credit balance for suburban customers is less than $3,200. The null hypothesis (H0) is that the mean is $3,200, and the alternative (Ha) suggests it is less than $3,200. The calculation of the t-statistic yielded a p-value below 0.05.

Thus, there is strong evidence that suburban customers have, on average, credit balances below $3,200, which could impact credit policies or marketing efforts aimed at this segment.

Confidence Intervals and Their Interpretation

For each of the variables examined, a 95% confidence interval was computed to provide a range of plausible values:

• Mean Income: The interval suggests that the true average income is likely above $45,000, reinforcing the hypothesis test result.
• Urban Population Proportion: The interval estimates the proportion of urban residents to be below 45%, corroborating the hypothesis test conclusion.
• Years in Home: The interval confirms that the average exceeds 8 years, aligned with the earlier test.
• Credit Balance for Suburban Customers: The interval indicates balances are probably below $3,200.

These intervals help quantify the uncertainty around the sample estimates, providing a range where the true population parameters are likely to fall with 95% confidence.

Conclusion and Managerial Implications

The statistical analyses provide compelling evidence supporting the initial beliefs in most cases. The average income surpasses $45,000, the proportion of urban customers is under 45%, customers tend to stay in their homes for more than 8 years, and suburban customers have relatively low credit balances below $3,200. These insights can guide strategic decisions such as targeted marketing, credit policies, and resource allocation. Communicating these findings in simple terms underscores that the company's customers are generally affluent and stable, with distinct demographic patterns that can be exploited for business growth.

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