Question

Consider the following 10 observations on the lifetime (in hours) for a certain type of component: \(152.7 .\) \(172.0,172.5,173.3,193.0,204.7,216.5,234.9,262.6, 422.6\). Construct a normal probability plot, and comment on the plausibility of a normal distribution as a model for component lifetime.

Short answer

Creating the Normal Probability Plot involves understanding the concept, calculating percentiles for the given data, plotting the values, creating the plot, and finally interpreting the plot. The interpretation is subjective and is based on the degree to which the points form a straight line.

Step-by-step solution

Understanding Normal Distribution Plots

In a normal probability plot or normal quantile plot, data from the distribution in question versus data from the standard normal distribution are plotted. When the data points of these two data sets form an approximately straight line, the distribution of the data set in question can be considered to be close to normal.

Calculating Percentiles

To construct this plot, percentiles must be calculated for each number in the set. This will be done using the formula \[ percentile = 100 * (i - 0.5) / n \] , where \(i\) is the rank of the specific number in the ordered set of data, and \(n\) is the total number of data. All ten observations must be ordered, and then for each observation, apply the aforementioned formula to calculate the percentile.

Constructing Normal Probability Plot

When the percentile for each observation has been calculated, plot them with the lifetime observation values. The lifetime will be on the x-axis, and the percentile on the y-axis. Connect them in order with a line to form the plot.

Interpret the Normal Probability Plot

The data can be considered to be approximately normally distributed if the points fall approximately along a straight line in the plot. A substantial departure from a straight line suggests a departure from normality.

Key concepts

Normal Distribution

The normal distribution is a bell-shaped curve that is symmetric around the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In real-world situations, many things roughly follow a normal distribution, such as heights of people, test scores, or errors in measurements.

When we say data is 'normally distributed,' we refer to a tendency to cluster around the central value, with the probabilities for values further away from the mean tapering off equally in both directions. Mathematical properties are highly desirable when working with statistics, as they facilitate easier calculation of probabilities and have well-understood behavior.

In the context of the exercise, considering the lifetime of components, if their lifetimes are normally distributed, we should expect most components to have a lifespan close to the average, with fewer and fewer lasting significantly shorter or longer.

Percentiles in Statistics

Percentiles are measures that divide a set of data into 100 equal parts, giving insights into the relative standing of a particular value within the data set. For example, if a student scores in the 75th percentile on a test, it means that 75% of students scored the same or less than them.

Percentiles are crucial in fields such as education, finance, and health because they offer a clear comparison among different data, indicating positions or ranks rather than actual performance. The formula provided in Step 2, \[ percentile = 100 * (i - 0.5) / n \], is used to find the percentile rank of each observation in a data set.

It's also important to note that the calculation of percentiles involves ranking the data points in ascending order, where each data point is then given its percentile value. This helps in constructing several types of plots — including normal probability plots — where these percentile ranks play a pivotal role.

Quantile Plot Interpretation

A quantile plot, also known as a normal probability plot, is a graphical tool to assess whether a data set follows a particular distribution, often the normal distribution. It plots the quantiles of the data set against the quantiles of the standard normal distribution.

When interpreting these plots, you are looking for a straight line that indicates the data follows the expected normal distribution. Any noticeable deviations from the straight-line pattern might suggest a different distribution is at play, and this is where a deep understanding of the plot can illuminate underlying characteristics of the data such as skewness, outliers, or heavy tails.

In the exercise's context, a normal probability plot helps determine the plausibility of assuming normality for component lifetimes. If the plots show a clear linear trend, the normal distribution could be a good model, otherwise, alternative distributions might have to be considered to create more accurate models or forecasts.