Question

Consider the following sample of 25 observations on the diameter \(x\) (in centimeters) of a disk used in a certain system: $$ \begin{array}{lllllll} 16.01 & 16.08 & 16.13 & 15.94 & 16.05 & 16.27 & 15.89 \\ 15.84 & 15.95 & 16.10 & 15.92 & 16.04 & 15.82 & 16.15 \\ 16.06 & 15.66 & 15.78 & 15.99 & 16.29 & 16.15 & 16.19 \\ 16.22 & 16.07 & 16.13 & 16.11 & & & \end{array} $$ The 13 largest normal scores for a sample of size 25 are \(1.965,1.524,1.263,1.067,0.905,0.764,0.637,0.519\) \(0.409,0.303,0.200,0.100\), and \(0 .\) The 12 smallest scores result from placing a negative sign in front of each of the given nonzero scores. Construct a normal probability plot. Does it appear plausible that disk diameter is normally distributed? Explain.

Short answer

After creating the normal probability plot, inspect it to determine whether the distribution appears to be roughly normal. If the points closely follow a straight line, then it is plausible that the distribution is normal. A non-linear pattern or systematic deviations from the line would suggest non-normality. The answer requires students to create and interpret such a plot.

Step-by-step solution

Data Arrangement

First, list the diameters in ascending order. This is important because the method for creating a normal probability plot is based on the principle of comparing our expected normal scores to the order statistics (i.e., the sorted values) from our sample.

Assign Normal Scores

Next, assign the given normal scores to sorted diameters. This means, the smallest diameter gets the smallest normal score (here -1.965), the next slightly larger diameter gets the next smallest normal score (here -1.524), and so on. Be sure to apply negative signs to the normal scores for the smallest half of your data. For the middle value, if the sample size is odd, assign a normal score of 0.

Construct the Plot

Draw a scatter plot. This plot should have the normal scores (z-scores) on the x-axis and the sorted diameters on the y-axis. If these points roughly form a straight line, then the assumption of normality is reasonable.

Interpret the Plot

Examine the plot. If the points closely follow a straight line, then the distribution of the data can be assumed to be normal. If the points do not closely follow a straight line, then the distribution of the data can be assumed not to be normal. Also, if there are any systematic deviations from the line (such as a curvature), this could indicate skewness or other forms of non-normality.

Key concepts

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a bell-shaped curve that is ubiquitous in statistics and is a fundamental concept in understanding how data behaves. At the core of its significance is the Central Limit Theorem. This theorem states that, under many circumstances, no matter what the initial distribution of a sample is, the distribution of the sample means will tend to look more and more like a normal distribution as the sample sizes increase.

When describing a normal distribution, two parameters are key - the mean (\(\bar{x}\)) and the standard deviation (\( \text{SD} \)). The mean indicates where the center of the bell curve is located on the number line, while standard deviation measures the spread of the data around the mean. A perfectly normal distribution is symmetric around the mean, with exactly half of the values falling below the mean and half above it. Real-world data often attempts to model this behavior, which is why the normality of a dataset is frequently assumed in statistical testing and process control.

Z-scores

A z-score, also known as a standard score, is the number of standard deviations by which a data point is above or below the mean of the data set. It's a way to compare scores from different scales, or different datasets, by normalizing the data. This is expressed by the formula: \[ z = \frac{(x - \bar{x})}{\text{SD}} \] Where \(x\) is a single data point, \( \bar{x}\) is the mean and SD is the standard deviation of the data set. Z-scores allow us to describe any data point in terms of its relationship to the mean and standard deviation of the dataset it belongs to.

A z-score of 0 indicates the value is exactly the mean, while a positive or negative z-score indicates the number of standard deviations the value is above or below the mean, respectively. In the context of a normal probability plot, we use these scores to determine how well the data fits the pattern of a normal distribution.

Data Arrangement

The arrangement of data is a crucial step in graphical analyses like constructing a normal probability plot. Organizing data systematically helps with the visualization and interpretation of the relationship, if any, between the observed values and the theoretical expectations.

The usual first step is to sort the data in ascending order, which aids in the identification of trends, such as increasing or decreasing patterns. For a normal probability plot, each data point is aligned with its expected position in a normal distribution. The procedure involves calculating the expected z-scores for each order statistic (sorted data point) in the sample. This sorting and aligning process is foundational to assessing normality in a graphical manner, which is more intuitive for many than numerical analysis.

Scatter Plot

A scatter plot, or scattergraph, is one of the simplest and most powerful graphical displays for data. The plot consists of a collection of points, each representing a value from one variable determining the position on the X-axis, and the value from the other variable determining the position on the Y-axis. In the context of a normal probability plot, the scatter plot maps the sorted observations of the dataset against the expected z-scores of a normal distribution.

The x-axis typically displays the z-scores, while the y-axis shows the actual values from the dataset. If the data points follow roughly a straight line, the scatter plot suggests that the dataset might be normally distributed. This graphical method of evaluating the normality of a dataset is widely used because it is straightforward and visually intuitive. It's an excellent first check to assess the assumption of normality required for various statistical tests.