ISCAM: Investigating Statistical Concepts, Applications, and Methods

The response variable, whether a household is considered unbanked, is categorical and binary. Suppose we define "success" as "unbanked." Then we can let \(\pi_{white}\) - \(\pi_{Black}\) represent the difference in the population proportions that would be classified as unbanked.

\(H_0: \pi_{white}\) - \(\pi_{Black} = 0\) (there is no difference in the population proportions)

\(H_a: \pi_{white}\) - \(\pi_{Black} \neq 0\) (there is a difference)

Note: The research question was phrased in terms of "whether there is a difference" so we will use a two-sided test.

The FDIC, like other government surveys, aims to gather a representative sample. Strategies such as weighting and combining results across surveys can also be utilized. In this case, we will treat the White and Black households as independent samples from their respective populations.

The sample sizes are large, so the normal distribution would be a reasonable model for the sampling distribution of the difference in sample proportions. Using this normal distribution model and assuming the null hypothesis is true, we would expect the distribution of the difference in sample proportions to be centered at zero, and our estimate of the standard deviation of this distribution is

\(\sqrt{0.0217(1-0.0217)(1/31653+1/4758)} \approx 0.0023\)

where 0.0217 = (447+342)/(31653 + 4758), the pooled estimate of the proportion of successes.

With this standard error, the observed difference in the sample proportions (0.0141 - 0.0719 = -0.058) is more than 25 standard errors from the hypothesized difference of zero. The probability of observing a standardized statistic at least this extreme (in either direction) by independent random sampling alone is essentially zero.

With such a small p-value (e.g., less than 0.01), we reject the null hypothesis. If the population proportions had been equal, this tells us that there is a very small probability of random sampling alone leading to sample proportions at least as far apart as those found in the FDIC survey. Therefore, we have strong evidence that the population proportion who are unbanked is not the same among Black and White households. A 95% confidence interval for the difference in the population proportions is (-0.0652, -0.0503). So we are 95% confident that percentage of White households is about 5 to 6.5 percentage points lower than the percent of Black households in the U.S. This is not a large difference in a practical sense, but we don’t believe a difference in sample proportions this large could have arisen by random sampling alone (in particular due to the large sample sizes). Note that the imbalance in the two sample sizes really does not negatively impact our comparison of the two proportions.

Because the combined sample size is much smaller between these two groups and the two sample proportions are much closer together, both of these factors would indicate a larger p-value. The p-value for the two-sample z-test does increase to 0.0161, but is still statistically significant even though the difference is now only about 1 percentage point.

For households that have a household income below $15,000 and a Black householder, about 15.9% of such households in this sample were unbanked. This is a much higher rate than we saw overall.

The difference between these two sample proportions is a bit larger (about 11 percentage points), but the sample sizes are also much smaller. The smaller sample sizes will increase the p-value. The z-value is indeed lower (\(z\) = 8.90), but still highly significant.

In the higher income range, the proportion unbanked is much smaller. If most of the respondents are in this higher range, this pulls down the overall percentage. The p-value is not necessarily impacted much by the lower proportions, the key is the difference in proportions. Even though this is likely smaller when comparing smaller numbers, the standard error will also decrease as the overall proportion is closer to 0 than 0.50.

In the two income groups we examined, we do see a consistently higher rate of unbanked households among Black and Hispanic households compared to White households. We also notice that the unbanked rate is similar (and even changes direction) between the Black and Hispanic households between income categories.

This is where a relative risk calculation might be more meaningful as the Hispanic proportion is 1.6 times larger than the Black proportion.

The missing values do decrease the sample sizes quite a bit, which we have seen directly impacts the size of the p-value. But perhaps even more problematic, we don’t know whether those who don’t answer the income question are systematically different on any of the variables, like financial transactions, than those who do answer the question. This can cause the sample proportions to not accurately reflect the population proportions.

A key advantage of imputation is you do not have to lose a large proportion of your sample. However, it is difficult to check the assumptions behind imputation and whether you are mimicking data you have already collected and falsely making the sample appear more homogenous and less variable.