The 92 million Americans of age 50 and over control \(50 \%\) of all discretionary income \((A A R P \text { Bulletin, March } 2008) .\) AARP estimated that the average annual expenditure on restaurants and carryout food was \(\$ 1873\) for individuals in this age group. Suppose this estimate is based on a sample of 80 persons and that the sample standard deviation is \(\$ 550\) a. At \(95 \%\) confidence, what is the margin of error? b. What is the \(95 \%\) confidence interval for the population mean amount spent on restaurants and carryout food? c. What is your estimate of the total amount spent by Americans of age 50 and over on restaurants and carryout food? d. If the amount spent on restaurants and carryout food is skewed to the right, would you expect the median amount spent to be greater or less than \(\$ 1873 ?\)

Understanding the sample standard deviation is crucial when dealing with statistics. It represents the variability or dispersion of a sample set of data. In simple terms, if the data points are spread out over a large range of values, you'll have a higher standard deviation, and if they are more clustered, the standard deviation will be lower.

When we say that the sample standard deviation for the average annual expenditure on restaurants and carryout food is \$550\$$, it implies that the spending amounts of individuals in the sample vary on average by this amount from the sample mean. It's important to note that the sample standard deviation can give us insights into the predictability and stability of data. A higher standard deviation in this context might suggest that spending habits are quite diverse among the sampled population.

The margin of error is another key concept in confidence interval estimation. It provides a range within which we can be sure that the population mean lies, to a certain level of confidence. It's a way of acknowledging that there's always some uncertainty when we're sampling.

For example, in our exercise, we determined the margin of error to be approximately \$120.42\$$. This number tells us that, while we have a sample mean expenditure of \$1873\$$, we acknowledge that the true population mean could reasonably be as low as \$1752.58\$$ or as high as \$1993.42\$$. It’s calculated by multiplying the critical value (determined from the confidence level) with the standard error of the sample mean. This margin of error is essential for researchers and businesses to understand the potential range of an estimate and its precision.

The population mean, in the context of our example on expenditures, represents the average amount spent by the entire population of Americans aged 50 and over on restaurants and carryout food. It is the parameter we aim to estimate by using our sample data.

As we work with samples, direct measurement of the population mean is often impractical or impossible due to large population sizes. Therefore, we use the sample mean as an estimate. In this case, \$1873\$$ represents the sample mean, which is our best guess for the population mean. However, because our sample is not the whole population, our estimate comes with a certain level of uncertainty, which is where the confidence interval becomes valuable, providing a range rather than a fixed number for the population mean.

The confidence level is a measure of certainty regarding how accurately a sample reflects the population from which it is drawn. It's typically expressed as a percentage, such as 95%, and indicates that if we were to take many samples and calculate confidence intervals for each of them, we would expect approximately 95% of those intervals to contain the true population mean.

In our exercise problem, a 95% confidence level means we can be 95% confident that the interval from \$1752.58\$$ to \$1993.42\$$ includes the true mean annual expenditure on restaurants and carryout food for Americans aged 50 and over. It reflects how confident we are in our sampling process and analysis. However, it's essential to understand that an increase in confidence level requires a broader margin of error, which results in a wider confidence interval.