Problem 29 - Basic Statistics for Business and Economics

Question

A chain of sport shops catering to beginning skiers, headquartered in Aspen, Colorado, plans to conduct a study of how much a beginning skier spends on his or her initial purchase of equipment and supplies. Based on these figures, they want to explore the possibility of offering combinations, such as a pair of boots and a pair of skis, to induce customers to buy more. A sample of their cash register receipts revealed these initial purchases: $$ \begin{array}{rrrrrrrrr} \hline \$ 140 & \$ 82 & \$ 265 & \$ 168 & \$ 90 & \$ 114 & \$ 172 & \$ 230 & \$ 142 \\ 86 & 125 & 235 & 212 & 171 & 149 & 156 & 162 & 118 \\ 139 & 149 & 132 & 105 & 162 & 126 & 216 & 195 & 127 \\ 161 & 135 & 172 & 220 & 229 & 129 & 87 & 128 & 126 \\ 175 & 127 & 149 & 126 & 121 & 118 & 172 & 126 & \\ \hline \end{array} $$ a. Arrive at a suggested class interval. Use five classes, and let the lower limit of the first class be \(\$ 80\). b. What would be a better class interval? c. Organize the data into a frequency distribution using a lower limit of \(\$ 80\). d. Interpret your findings.

Step-by-Step Solution

Answer

Use a class interval of $37, with most purchases between $117 and $153.

Step 1: Determine the Range of the Data

First, find the range of the data set by subtracting the smallest value from the largest value. The smallest value is $82 and the largest is $265. Thus, the range is $265 - $82 = $183.

Step 2: Calculate Class Interval

To find the class interval, divide the range by the number of classes requested, which is 5. Thus, the class interval is \(\frac{183}{5} = 36.6\). Rounding up gives a class interval of 37.

Step 3: Construct Class Limits

With a lower limit of $80 for the first class and a class interval of 37, the class limits are as follows: - Class 1: $80 to $116 - Class 2: $117 to $153 - Class 3: $154 to $190 - Class 4: $191 to $227 - Class 5: $228 to $264

Step 4: Create the Frequency Distribution Table

Tally the data within these classes to form the frequency distribution: - Class 1 ($80 to $116): 6 - Class 2 ($117 to $153): 15 - Class 3 ($154 to $190): 13 - Class 4 ($191 to $227): 6 - Class 5 ($228 to $264): 3

Step 5: Interpret the Frequency Distribution

From the frequency distribution, the most purchases are in the range of $117 to $153, indicating a potential target group for the company's marketing strategy of offering combination deals.

Key Concepts

Class Interval

When you are working with data, organizing it into a frequency distribution can give you a clearer picture of what's going on. The class interval plays a pivotal role in this setup. It refers to the range of values that each group, or "class," can cover in the frequency distribution. Think of it like splitting a bookshelf into sections where each section holds a particular type of book. In our exercise, we decided on using five classes.
Imagine the class interval as the width of each section. To determine this, you begin with the range of the data—essentially, how "wide" or varied the data is—then divide by the number of classes you want.
For instance, a calculated class interval of 36.6 means each group within our data should be 37 units wide once rounded up.

• This rounding is key because class intervals should cover a whole number of data points without decimal excess.
• Each class captures data inclusively from one endpoint to the next, without overlaps.

Range of Data

Data can tell stories of varying expanses, and the range of data expresses just how far apart these values can stretch. Calculating it is straightforward; simply subtract the smallest number from the biggest one within your dataset.
In our example, the smallest value was $82 while the largest reached $265, resulting in a range of $183. But why is this useful? Knowing the range helps in understanding the overall spread of data, which assists in deciding how to group this information efficiently.

• Without this, grouping values might either squish all your data together too tightly or spread it out too thinly.
• It's a crucial first step before determining other elements like class intervals or limits.

This foundational step often sets the tone for how other calculations and insights will be derived in the dataset.

Data Interpretation

Once the frequency distribution is organized, interpreting the data becomes the next big task. It's not just about numbers; it's about what these figures are revealing. In our task, after setting up the frequency distribution, a quick look shows the majority of purchases falling within the $117 to $153 range.
This reveals potential insights such as which price range customers are most comfortable with or when they might be more inclined to make purchases.

• Such interpretations can directly influence marketing strategies, like focusing promotions on popular price ranges.
• The concentration of purchases at these levels might be a hint towards the types of products that are popular.

Data interpretation thus bridges the gap between raw figures and strategic decisions, transforming numbers into actionable insights.

Class Limits

Setting class limits is a critical part of forming an understandable frequency distribution. These limits act as the borders that define where each class starts and ends.
In our case, with a starting point of $80 and a class interval of 37, each subsequent class is built upon this initial limit.

• Class 1 span from $80 and ends at $116.
• Class 2 goes from $117 to $153, and so on.

The upper limit of one class will always border the lower limit of another without any gap or overlap. Proper class limits ensure that each item of data can be neatly categorized into one and only one class, providing clarity and precision to how the data is being viewed.
Just like reading a map, clear demarcation helps avoid confusion about where one territory ends and the next begins.