Question

Increases in worker injuries and disability claims have prompted renewed interest in workplace design and regulation. As one particular aspect of this, employees required to do regular lifting should not have to handle unsafe loads. The article "Anthropometric, Muscle Strength, and Spinal Mobility Characteristics as Predictors of the Rating of Acceptable Loads in Parcel Sorting" (Ergonomics [1992]: 1033-1044) reported on a study involving a random sample of \(n=18\) male postal workers. The sample mean rating of acceptable load attained with a work-simulating test was found to be \(\bar{x}=9.7 \mathrm{~kg}\). and the sample standard deviation was \(s=4.3 \mathrm{~kg}\). Suppose that in the population of all male postal workers, the distribution of rating of acceptable load can be modeled approximately using a normal distribution with mean value \(\mu\). Construct and interpret a \(95 \%\) confidence interval for \(\mu\).

Short answer

The 95% confidence interval for \(\mu\) is calculated using the above steps based on the given sample. The interpretation would depend on the calculated range which is a result of these computations.

Step-by-step solution

Define and Compute the Standard Error

To begin, we first need to calculate the standard error. The standard error of the mean is given by the formula \(SE = \frac{s}{\sqrt{n}}\), where \(s\) is the standard deviation and \(n\) is the number of observations in the sample. Here, our standard deviation, \(s\), is 4.3 kg, and the number of samples, \(n\), is 18, so our standard error is \(SE = \frac{4.3}{\sqrt{18}}\).

Find the z-value

For a 95% confidence interval, we look for the z-value that leaves 5% of the probability distribution in the tails, split evenly at 2.5% in each tail. This z-value can be looked up in standard statistical tables, or be found using appropriate functions in statistical software. The z-value corresponding to a 95% confidence interval is about 1.96.

Calculate the Confidence Interval

The confidence interval is calculated using the formula \(CI = \bar{x} \pm z * SE\), where \(\bar{x}\) is mean of the sample, \(z\) is our z-value for the chosen level of confidence, and \(SE\) is the standard error. Here, our sample mean, \(\bar{x}\), is 9.7 kg, and our standard error has been computed in the first step, and z-value has been obtained in the second step. Substituting these values into the formula will yield the confidence interval for \(\mu\).

Interpret the Confidence Interval

The resulting confidence interval can be interpreted as follows: We are 95% confident that the true mean value for the acceptable load for all male postal workers is within the calculated range.

Key concepts

Standard Error

The standard error (SE) is a statistical measure that quantifies the amount of variation or dispersion of a sample mean relative to the true population mean. It's a crucial concept in statistics as it helps researchers understand how far off a sample's mean is likely to be from the actual mean of the entire population.

To compute the standard error, you need the sample standard deviation, denoted as 's', and the sample size, denoted as 'n'. The formula is given by:
\[\begin{equation} SE = \frac{s}{\sqrt{n}} \end{equation}\]
In a real-world scenario, such as assessing the acceptable load ratings by postal workers, the standard error helps determine the reliability of the sample mean. A smaller standard error suggests a more reliable and precise sample mean, indicating that the sample is a good representation of the population.

Sample Mean

The sample mean, often represented by \(\bar{x}\), is the average value of a set of observations. It is calculated by adding up all the values and then dividing by the number of observations. When we have a sample mean, it’s important to remember we are looking at a representation of the broader population from which the sample was drawn.

To mathematically express the sample mean, you use the formula: \[\begin{equation} \bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i \end{equation}\]
where \(x_i\) represents each value in the sample and \(n\) is the total number of observations. This average can give us insights into the central tendency of the data.

In practical terms, like the study of postal workers and acceptable load weights, the sample mean gives us an estimate of what postal workers consider to be an acceptable load on average. Knowing the sample mean allows us to then apply it to conclusions about the general population of postal workers, including constructing confidence intervals to understand the variability of this estimate.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a symmetric, bell-shaped curve that represents the distribution of many types of data. Most values remain around the mean and the probabilities for values further away from the mean taper off equally in both directions.

Mathematically described by the equation of a bell curve, the main properties of a normal distribution include its mean \(\mu\), standard deviation \(\sigma\), and the total area under the curve representing the probability which is always 1. The standard normal distribution has a mean of 0 and a standard deviation of 1.

For practical applications like the one involving postal workers, assuming a normal distribution can simplify the process of making inferences about the population mean—since we can then use well-established statistical tools such as z-scores for constructing confidence intervals. With this approach, it is assumed that sample means will tend to form a normal distribution around the true population mean as the sample size increases, due to the Central Limit Theorem.