Older people often have a hard time finding work. AARP reported on the number of weeks it takes a worker aged 55 plus to find a job. The data on number of weeks spent searching for a job contained in the file JobSearch are consistent with the AARP findings \((A A R P\) Bulletin, April 2008). a. Provide a point estimate of the population mean number of weeks it takes a worker aged 55 plus to find a job. b. \(\quad\) At \(95 \%\) confidence, what is the margin of error? c. What is the \(95 \%\) confidence interval estimate of the mean? d. Discuss the degree of skewness found in the sample data. What suggestion would you make for a repeat of this study?

A point estimate provides a single value as an estimate of a population parameter. In the context of job search analysis for older workers, the point estimate represents an estimate of the population mean - that is, the average number of weeks all older workers might expect to spend looking for a job. To calculate this from a sample, we add all the weeks reported by individuals and divide by the number of respondents. This calculated mean, often denoted as \( \bar{X} \), is our point estimate. It's a valuable figure because it gives us a reference point for the central tendency of the dataset. Using point estimates allows researchers and policy makers to make informed guesses about the population from the sample data, though it is key to remember that a point estimate is susceptible to sampling error and does not reflect any measure of variability on its own.

The confidence interval adds context to the point estimate by providing a range within which we can say, with a certain level of confidence, that the true population parameter lies. For older workers' job search data, constructing a 95% confidence interval around the sample mean tells us that if we were to draw many samples and construct intervals in the same way, 95 out of 100 of those intervals would contain the actual average weeks it would take for older workers to find jobs.

The formula for this interval is the sample mean \( \bar{X} \) plus and minus the margin of error, with the margin defined as the critical value from the standard normal distribution (often 1.96 for 95% confidence) multiplied by the sample standard deviation divided by the square root of the sample size. This interval is crucial because rather than just estimating a single value, it acknowledges the possibility of variation and provides a plausible range for the population parameter.

Sample standard deviation (denoted as s) is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. When analyzing job search time for older workers, calculating the sample standard deviation is essential for two reasons: It helps us understand the variability of the job search duration among individuals in the sample, and it's vital for the calculation of both the margin of error and the confidence interval.

We calculate s by determining the variance, which involves subtracting the sample mean from each observation, squaring these differences, averaging them out, and finally taking the square root. Understanding the standard deviation gives context to the point estimate, as it identifies how much the individual data points are diverging from the average.

Data skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. In simple terms, skewness tells us if the data is evenly distributed around the mean, or if it's leaning towards the left (negative skew) or right (positive skew).

In the context of the job search for older workers, analyzing skewness is pivotal to interpret how evenly the job search duration is spread among the sample. Skewness can influence the interpretation of the mean; for example, a positively skewed distribution may have a mean that is higher than the median, which would indicate a presence of outliers on the high end (people taking a very long time to find a job).

It's important to assess skewness because significant skewness in data can impact the validity of the statistical analyses (like the construction of confidence intervals) that assume normal distribution. Researchers may need to consider data transformation or different analytical techniques to accommodate or understand the skewness in their data, especially for developing interventions or policy recommendations.