The significant decline of savings in the United States from the 1970s and 1980s
The significant decline of savings in the United States from the 1970s and 1980s to the 1990s and 2000s has been widely discussed by economists (money.cnn.com, June 30, 2010). According to the Bureau of Economic Analysis, the savings rate of American households, defined as a percentage of the disposable personal income, was 4.20% in 2009. The reported savings rate is not uniform across the country. A public policy institute conducts two of its own surveys to compute the savings rate in the Midwest. In the first survey, a sample of 160 households is taken and the average savings rate is found to be 4.48%. Another sample of 40 households finds an average savings rate of 4.60%. Assume that the population standard deviation is 1.4%. In a report, use the above information to: 1. Compute the probability of obtaining a sample mean that is at least as high as the one computed in each of the two surveys. 2. Use these probabilities to decide which of the two samples is likely to be more representative of the United States as a whole.
Analysis of Savings Rate Surveys in the Midwest
This report examines the statistical significance of two independent samples of household savings rates in the Midwest, conducting probability calculations to understand their representativeness relative to national trends. The analysis hinges on the assumption of a known population standard deviation and employs the standard normal distribution to evaluate the likelihood of observing the survey results given national savings behavior.
Statistical Foundations
The key parameters provided include the population standard deviation (σ = 1.4%) and the sample sizes (n1 = 160; n2 = 40), along with their respective sample means (x■1 = 4.48%; x■2 = 4.60%). For both samples, the goal is to determine the probability of obtaining a sample mean equal to or exceeding these observed values, assuming the true population mean corresponds to the national savings rate of 4.20%. This approach utilizes the standard normal distribution to compute the p-values associated with these sample means, enabling an assessment of their plausibility as representative samples.
Calculation of Probabilities for Sample Means
To compute the probability that the sample mean is at least as high as the observed mean, we calculate the z-score for each sample. The formula for the z-score is:
z = (x■ – µ) / (σ / √n)
where x■ is the sample mean, µ is the hypothesized population mean (4.20%), σ is the population standard deviation, and n is the sample size.
Sample 1: n=160, x■=4.48%
Calculating the z-score:
z■ = (4.48 – 4.20) / (1.4 / √160) ≈ 0.28 / (1.4 / 12.649) ≈ 0.28 / 0.1107 ≈ 2.53
The probability that the sample mean is at least 4.48% is the area to the right of z = 2.53 under the standard normal curve. Consulting standard normal distribution tables or using statistical software, we find:
P(Z ≥ 2.53) ≈ 0.0057
Sample 2: n=40, x■=4.60%
Calculating the z-score: z■ = (4.60 – 4.20) / (1.4 / √40) ≈ 0.40 / (1.4 / 6.3246) ≈ 0.40 / 0.2214 ≈ 1.81
Similarly, the probability that the sample mean is at least 4.60% is:
P(Z ≥ 1.81) ≈ 0.0351
Interpretation of Probabilities and Representativeness
The computed probabilities indicate that obtaining a sample mean of at least 4.48% in the first survey, assuming the true population mean is 4.20%, is approximately 0.57%. Conversely, the probability for a sample mean of at least 4.60% in the second survey is approximately 3.51%. Since the probability associated with the first sample is lower, it suggests that observing such a high average savings rate in the larger sample is less likely if the true population mean is 4.20%. Therefore, the second sample’s observed mean is comparatively more plausible under the assumption that the national savings rate remains around 4.20%. This implies that the second sample may be more representative of the national savings behavior.
Conclusion
In conclusion, statistical evidence points toward the second survey (with an average savings rate of 4.60%) being more aligned with the broader national trend. The higher probability of observing such a mean suggests that this smaller sample better captures the true variance and characteristics of the American
household savings rate during the period under analysis. However, the low overall probabilities also underscore that both observed sample means are somewhat higher than the national average, which could reflect regional differences or sampling variability. Overall, these findings highlight the importance of sample size, variability, and the use of statistical measures when evaluating the representativeness of survey data in economic research.
References
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