Question

Consider the following 10 observations on the lifetime (in hours) for a certain type of power supply: $$ \begin{array}{lllll} 152.7 & 172.0 & 172.5 & 173.3 & 193.0 \\ 204.7 & 216.5 & 234.9 & 262.6 & 422.6 \end{array} $$ Construct a normal probability plot, and comment on whether it is reasonable to think that the distribution of power supply lifetime is approximately normal. (The normal scores for a sample of size 10 are -1.539,-1.001,-0.656 , \(-0.376,-0.123,0.123,0.376,0.656,1.001,\) and \(1.539 .)\)

Short answer

Whether the power supply runtime data is normally distributed can't be determined definitively without constructing and examining the normal probability plot. This process involves sorting the data, pairing it with the provided normal scores, plotting each data point, and then visually inspecting the resulting plot.

Step-by-step solution

Arrange the Data

First, arrange the observations in ascending order. The result is 152.7, 172.0, 172.5, 173.3, 193.0, 204.7, 216.5, 234.9, 262.6, and 422.6.

Compare with Normal Scores

Compare each observation with the provided normal scores. This can be done by pairing each sorted observation with each of the normal scores -1.539, -1.001, -0.656, -0.376, -0.123, 0.123, 0.376, 0.656, 1.001, and 1.539.

Create a Scatter Plot

Plot the literacy data points, or observations, on a graph against their corresponding normal scores. This plot is called a normal probability plot or Q-Q plot.

Interpret the Scatter Plot

Visually inspect the scatter plot. If the observations fall approximately in a straight line, then it can be concluded that the data follows a normal distribution. If the observations do not fall along a straight line, then the data does not follow a normal distribution.

Key concepts

Normal Distribution

The concept of a normal distribution, also known as a Gaussian distribution, is central to many areas of statistical analysis. It represents a continuous probability distribution characterized by a symmetric, bell-shaped curve. Most importantly, the mean, median, and mode of a normal distribution are all equal and located at the center of the curve.

The relevance of the normal distribution arises from the Central Limit Theorem, which states that the sum of many independent random variables will tend to be normally distributed, regardless of the original distribution of the variables. This makes it critically important for interpreting sample data and understanding the behavior of averages in the context of the larger population.

In our exercise dealing with power supply lifetimes, if the data is normally distributed, we would expect the values to conform to this bell-shaped curve when graphed, with most observations clustering around the mean and fewer as they move away from it.

Q-Q Plot

A Q-Q plot, or quantile-quantile plot, is a graphical tool used to compare two probability distributions by plotting their quantiles against each other. If the two distributions being compared are similar, the points on the Q-Q plot will approximately lie on a straight line. In the context of our exercise, the Q-Q plot specifically helps us determine if our dataset on power supply lifetimes follows a normal distribution.

To create a Q-Q plot, we must first calculate or obtain the theoretical quantiles from a normal distribution, which are often referred to as 'normal scores'. Then, we pair each observational data point with its corresponding theoretical quantile. By plotting these pairs on a graph, we visualize how closely our data follows a normal distribution. Any deviations from the straight line indicate departures from normality.

Statistical Analysis

Statistical analysis encompasses a wide range of methods for exploring, describing, and inferencing from data. It is an essential process in making sense of data collected from experiments, surveys, and observational studies. Core to statistical analysis is the concept of distribution, which describes how often each value occurs, or its frequency.

Key procedures include summarizing data patterns with measures of central tendency (like the mean and median) and spread (like variance and standard deviation), hypothesis testing, regression analysis, and creating confidence intervals. In this exercise, we are primarily concerned with descriptive statistics and the potential normality of our dataset, which would affect the inferences we could make about the lifetimes of power supplies.

Data Visualization

Data visualization is the presentation of data in a pictorial or graphical format. It enables users to see analytics presented visually, so they can grasp difficult concepts or identify new patterns easier. Effective visualizations help users analyze and reason about data and evidence. It makes complex data more accessible, understandable, and usable.

In our exercise, the normal probability plot is a form of data visualization. By plotting the sorted observations against their expected score on a normal distribution, we create a visual representation of how well the data align with a normal distribution. This intuitive understanding is invaluable because it can often reveal subtleties in the data that raw numbers or simple statistics might not make apparent.