The significant decline of savings in the United States from

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: