Problem 5 - Basic Statistics for Business and Economics

Question

Wachesaw Manufacturing, Inc. produced the following number of units the last 16 days. $$ \begin{array}{|llllllll|} \hline 27 & 27 & 27 & 28 & 27 & 25 & 25 & 28 \\ 26 & 28 & 26 & 28 & 31 & 30 & 26 & 26 \\ \hline \end{array} $$ The information is to be organized into a frequency distribution. a. How many classes would you recommend? b. What class interval would you suggest? c. What lower limit would you recommend for the first class? d. Organize the information into a frequency distribution and determine the relative frequency distribution. e. Comment on the shape of the distribution.

Step-by-Step Solution

Answer

Use 5 classes with an interval of 2 starting at 25. Distribution is left-skewed.

Step 1: Identify the Range

First, we need to identify the range of the data, which is the difference between the maximum and minimum values. From the given data, the maximum value is 31 and the minimum value is 25. Thus, the range is: \( 31 - 25 = 6 \).

Step 2: Determine the Number of Classes

Using Sturges' Rule, a common guideline for determining the number of classes, we calculate the number of classes as follows: \( k = 1 + 3.322 \log_{10}(n) \), where \( n \) is the sample size. For 16 units: \[ k = 1 + 3.322 \log_{10}(16) \approx 1 + 3.322 \times 1.204 \approx 5. \] So, we recommend 5 classes.

Step 3: Calculate Class Interval

The class interval is determined by dividing the range by the number of classes and rounding up if necessary. Thus, class interval \( C \) is: \[ C = \lceil \frac{6}{5} \rceil = 2. \] Hence, the recommended class interval is 2.

Step 4: Choose the Lower Limit for the First Class

A sensible lower limit for the first class is the minimum value or close to it. Given our data set, 25 is a good starting point for the first class lower limit.

Step 5: Organize Data into a Frequency Distribution

We now set up the classes starting at 25 with a width of 2. - 25-26 - 27-28 - 29-30 - 31-32 Counting entries: - 25-26: 6 entries (25, 25, 26, 26, 26, 26) - 27-28: 7 entries (27, 27, 27, 28, 27, 28, 28) - 29-30: 2 entries (30) - 31-32: 1 entry (31)

Step 6: Calculate Relative Frequency

Divide the frequency of each class by the total number of observations (16) to get the relative frequency:- 25-26: \( \frac{6}{16} = 0.375 \)- 27-28: \( \frac{7}{16} = 0.4375 \)- 29-30: \( \frac{2}{16} = 0.125 \)- 31-32: \( \frac{1}{16} = 0.0625 \)

Step 7: Comment on the Shape of the Distribution

The distribution is skewed towards the lower values since most of the observations fall within the earlier classes (25-26 and 27-28). There is also one outlier class, 31-32, indicating a single high observation compared to others.

Key Concepts

Class Interval

In frequency distribution, a class interval is crucial for organizing data into groups or classes. These intervals help us to simplify and make sense of the data. Think of it like shelving books in a library; we arrange similar subjects together for easy access. The class interval refers to the numerical span between one category in our data set and the next.

Firstly, to determine the class interval, we need to have the range of the data. The range is the difference between the highest and lowest number in your dataset. Once you have the range, the class interval is typically calculated by dividing the range by the number of classes you wish to have. Rounding this number up sometimes ensures that the data fits nicely into classes. In our example, the range was 6 and dividing this by the number of classes (5) gave us a class interval of 2.

This means each class covers two units, such as 25-26 and 27-28. A consistent class interval allows for smoother data comparisons across different classes when looking at distributions.

Relative Frequency

Relative frequency provides insights into how often a specific event or data point appears in a dataset relative to the total number of data points. It is expressed as a fraction or a percentage, representing the likelihood or proportion of that specific data point among the entire dataset.

The calculation of relative frequency involves dividing the count of observations in a particular class by the total number of observations. This helps in understanding the prominence of certain classes within the dataset. In the context of our exercise, we looked at how many data points fell into each specified class, like 25-26 and 27-28, and then divided those counts by the total number of observations, which is 16.

For example, the relative frequency of the class 25-26 was 0.375. This is because 6 out of 16 data points fell in this range, making up 37.5% of the dataset. By examining the relative frequencies, we gain a better understanding of the data's distribution and which values are more prevalent.

Sturges' Rule

Sturges' Rule is a guideline for determining the optimal number of classes to use in a frequency distribution. Named after the statistician Herbert Sturges, it's a formula that balances precision and simplicity when summarizing data.

The rule suggests a formula: \( k = 1 + 3.322 \log_{10}(n) \), where \( k \) is the number of classes and \( n \) is the number of data points. It leverages logarithms to ensure that the number of classes grows at a reasonable rate compared to the size of your dataset, without overwhelming the analysis with too many or too few classes.

In our exercise, with a sample size of 16, Sturges’ Rule recommended about 5 classes. This helped us to segment the data efficiently and ensured that each class had a significant amount of data points to make meaningful analysis possible. Understanding how many classes to use helps in creating a clearer picture of the data's overall trends and distributions without losing crucial details.