91Ó°ÊÓ

A stem-and-leaf plot is a useful tool in descriptive statistics, especially when organizing smaller datasets. It essentially summarizes data by organizing them into 'stems' and 'leaves'. In the context of the provided exercise, each number representing the life expectancy of a fuel pump has a stem, which is the integer part before the decimal, and a leaf, which is the decimal part to the right of the decimal point.

To construct the plot, start by writing down each unique integer part (or stem) in a column. Then, next to each integer, list the corresponding decimal parts (leaves) for numbers that share the same integer. This organized format allows us to quickly observe the distribution and frequency of the data without losing the actual data values. For instance, a plot with a stem of '0' and leaves '2, 3, 2, 3, 7, 2, 3, 5' would show the decimal parts of numbers that start with 0.

Stem-and-leaf plots not only help in displaying the shape of the data distribution but also retain the original data values, making it easier to find specific data points if needed.

Relative frequency distribution is a way to show how often each category or class of data occurs relative to the total number of observations. It’s expressed as a percentage or a fraction. For example, in the exercise with fuel pumps, every number with the same integer digit is grouped together – these groups make up the classes.

To calculate the relative frequency of each class, divide its count by the total number of observations and multiply by 100 to convert it to a percentage. Suppose we had eight observations of fuel pumps with a lifespan starting with 0, this would mean the relative frequency for the 0.x class is \( \frac{8}{39} \times 100 \approx 20.5\% \).

This method provides a clear picture of the distribution of data. It helps us understand which ranges of fuel pump lifespans are more common, providing insight into typical lifespans and highlighting any unusual values.

The sample mean is a foundational concept in statistics used to find the average value of a sample data set. Calculating it involves adding up all individual data values and then dividing by the total number of observations. In this exercise, the sample mean tells us about the average lifespan of the fuel pumps.

To compute it: sum all the lifespan values of the fuel pumps and then divide by the total count, which is 39 in this case. Mathematically, it’s represented as \( \bar{x} = \frac{\sum x_i}{n} \), where \( \sum x_i \) is the sum of all data points and \( n \) is the number of data points.

This straightforward calculation offers significant insight into the general performance of the fuel pumps, allowing us to gauge how they’re likely to perform over time.

The sample range provides a simple measure of the spread between the smallest and largest values in a data set. It’s easy to calculate—just subtract the smallest observation from the largest. In our context, it reveals the gap between the shortest and longest lifespans of the fuel pumps.

Calculating the range involves identifying the minimum and maximum values within the data set. Suppose the smallest value is 0.2 years and the largest is 6.5 years. The range, therefore, is \( 6.5 - 0.2 = 6.3 \) years.

While the range gives insight into variability, it’s important to remember that it can be affected by outliers, or extreme values, and does not necessarily represent the overall spread of the data well if they are present.

The sample standard deviation is a crucial measure of how spread out the observations in a sample are around their mean. Unlike the range, which depends only on the extreme ends, the standard deviation uses every data point to determine variability.

To calculate it, several steps are involved: first, find the mean of the data set. Next, subtract the mean from each observation to find their deviations, square these deviations, and then calculate the average of these squared deviations. Finally, take the square root of that average to find the standard deviation.
Mathematically, it is represented as \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \), where \( x_i \) is each observation, \( \bar{x} \) is the sample mean, and \( n \) is the total number of observations.

This calculation gives insight into how much the individual fuel pump lifespans deviate from the average, revealing more about their reliability and consistency in performance over time. By knowing this, we can better understand how predictable the pump lifespans are.