Sample Of 250 Households In NY Showed 62 Paid In 2004

In 2004 A Sample Of 250 Household In Ny Showed That 62 Paid Their Mon

In 2004, a sample of 250 households in NY showed that 62 paid their monthly phone bills by debit card. In 2009, a sample of 350 households in the same city showed that 110 paid their monthly phone bill by debit card. Based on the sample data, can we conclude that there is a difference between the population proportion of 2004 households and 2009 households in the same city?

a. Determine if the sample sizes are large enough.

b. Conduct the appropriate hypothesis test. Report the p-value. Would the null hypothesis be rejected?

c. Calculate the pooled estimate for overall proportion.

Paper For Above instruction

Introduction

The growth of digital banking and payment methods has transformed financial transactions over the past decades. Specifically, understanding the change in the population proportion of households using debit cards for utility payments helps financial institutions, policymakers, and service providers assess trends and make informed decisions. This paper aims to analyze whether there has been a significant change between 2004 and 2009 in the proportion of households in NY paying their monthly phone bills via debit card, using hypothesis testing methods.

Assessing Sample Size Adequacy

Before conducting a hypothesis test for difference in proportions, it is crucial to verify whether the sample sizes are sufficiently large for the approximation to the normal distribution to be valid. The rule of thumb states that both np and n(1-p) should be at least 5, where n is the sample size, and p is the sample proportion.

For 2004:

- Sample size (n1) = 250

- Number paying by debit card = 62

- Sample proportion (p1̂) = 62/250 = 0.248

Verify:

np = 250 * 0.248 = 62 (which is greater than 5)

n(1-p) = 250 * 0.752 = 188 (which is greater than 5)

For 2009:

- Sample size (n2) = 350

- Number paying by debit card = 110

- Sample proportion (p2̂) = 110/350 ≈ 0.314

Verify:

np = 350 * 0.314 ≈ 110, which is greater than 5

n(1-p) = 350 * 0.686 ≈ 240, which is greater than 5

Since both conditions are satisfied for both samples, the sample sizes are adequately large for the normal approximation and hypothesis testing can proceed.

Hypothesis Testing Framework

The hypothesis test for comparing two population proportions is formulated as:

- Null hypothesis (H0): p1 = p2 (no difference in proportions)

- Alternative hypothesis (Ha): p1 ≠ p2 (there is a difference)

Using sample data:

- p1̂ = 0.248

- p2̂ = 0.314

- n1 = 250

- n2 = 350

We conduct a two-tailed Z-test for difference in proportions.

Calculating the Pooled Proportion

The pooled proportion (p̂) combines the data from both samples under the assumption that H0 is true:

\[

p̂ = \frac{\text{Total number paying by debit}}{\text{Total sample size}} = \frac{62 + 110}{250 + 350} = \frac{172}{600} ≈ 0.287

\]

Test Statistic Calculation

The standard error (SE) for the difference in proportions is:

\[

SE = \sqrt{p̂ (1 - p̂) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}

\]

\[

SE = \sqrt{0.287 \times 0.713 \left( \frac{1}{250} + \frac{1}{350} \right)} \approx \sqrt{0.204 \times (0.004 + 0.00286)} \approx \sqrt{0.204 \times 0.00686} \approx \sqrt{0.001404} \approx 0.0375

\]

The Z-value is:

\[

Z = \frac{p_{1̂} - p_{2̂}}{SE} = \frac{0.248 - 0.314}{0.0375} = \frac{-0.066}{0.0375} \approx -1.76

\]

P-Value and Conclusion

Using standard normal distribution tables or statistical software:

- P-value for Z ≈ -1.76 (two-tailed test): 2 P(Z 0.039 = 0.078

Since the p-value (~0.078) exceeds the conventional significance level (α=0.05), we fail to reject the null hypothesis.

Interpretation: There is insufficient evidence at the 5% significance level to conclude a significant difference in the population proportions of households paying their phone bills via debit card between 2004 and 2009 in NY.

Conclusion

The analysis indicates that, although the sample proportions changed from 0.248 to 0.314, this difference was not statistically significant at the 5% level. The sample sizes were adequate, and the hypothesis test suggests that the observed change could be due to sampling variability rather than a true population difference. These findings imply that the shift in debit card usage for phone bills among NY households, during this period, was modest and not statistically conclusive.

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