91Ó°ÊÓ

The data on the status of 50 students given in Table \(2.2\) of Section \(2.1\) are reproduced here. $$ \begin{array}{llllllllll} \text { J } & \text { F } & \text { SO } & \text { SE } & \text { J } & \text { J } & \text { SE } & \text { J } & \text { J } & \text { J } \\ \text { F } & \text { F } & \text { J } & \text { F } & \text { F } & \text { F } & \text { SE } & \text { SO } & \text { SE } & \text { J } \\ \text { J } & \text { F } & \text { SE } & \text { SO } & \text { SO } & \text { F } & \text { J } & \text { F } & \text { SE } & \text { SE } \\ \text { SO } & \text { SE } & \text { J } & \text { SO } & \text { SO } & \text { J } & \text { J } & \text { SO } & \text { F } & \text { SO } \\ \text { SE } & \text { SE } & \text { F } & \text { SE } & \text { J } & \text { SO } & \text { F } & \text { J } & \text { SO } & \text { SO } \end{array} $$ a. Prepare a frequency distribution table. b. Calculate the relative frequencies and percentages for all categories. c. What percentage of these students are juniors or seniors? d. Draw a bar graph for the frequency distribution.

Step by step solution

01

Step 1

Count how many times each status appears in the array. It'll form the frequency distribution table for this data set.

02

Step 2

Calculate the relative frequencies and percentages for all categories. This can be done using the following formulas: \[ \text { Relative frequency of a category }=\frac {\text { frequency of the category }}{\text { total number of data points }} \] \[ \text { Percentage of a category }= (\text { Relative frequency of the category }) \times 100 \]

03

Step 3

To find out the percentage of students who are juniors or seniors, add the percentages calculated for juniors (J) and seniors (SE).

04

Step 4

Prepare a bar graph for the frequency distribution. On the x-axis, place the different categories of students statuses. On the y-axis, place the frequency. The height of the bar for each category corresponds to its frequency in the dataset.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Relative Frequency

The concept of relative frequency is an essential part of understanding how often different categories appear within a dataset. To calculate the relative frequency, you first need to know the total number of data points, which in this case is 50 students. For each status category, count how many times it appears in the dataset. For example, if 'Juniors (J)' appears 20 times, calculate the relative frequency of juniors by dividing this frequency by the total number of students:

• Relative frequency of Juniors (J) = \( \frac{20}{50} = 0.4 \)

This will give you a relative frequency of 0.4 or 40%. Relative frequency helps us understand the proportion or share of each category compared to the entire dataset.Understanding relative frequencies allows you to compare categories within the data better and see which statuses are more common.

Bar Graph

Bar graphs are a visual way to display frequency data. They are particularly useful when you want to compare different categories side by side. To draw a bar graph for the frequency distribution of the student status data:

• First, place the different student categories (e.g., Juniors (J), Freshmen (F), Sophomores (SO), Seniors (SE)) on the x-axis.
• Next, label the y-axis with frequencies, which represent how many times each category appears in the dataset.

For each category, draw a vertical bar that reaches up to the corresponding frequency. For instance, if 'Sophomores (SO)' appears 10 times, draw a bar up to 10 on the y-axis. Bar graphs make it easy to see differences in frequency at a glance. The taller the bar, the higher the frequency, which makes spotting trends or patterns quicker and more straightforward.

Percentage Calculation

Percentage calculation is a handy tool to express data in a way that is easy to understand and interpret. Once you have the relative frequency of each category, converting it to a percentage is straightforward. You multiply the relative frequency by 100.

• Percentage of Juniors (J) = Relative frequency of Juniors (J) \( \times 100 = 0.4 \times 100 = 40\% \)

This method allows you to compare categories precisely. It’s particularly useful when trying to find out combined percentages, such as the total percentage of students who are either juniors or seniors.To find this, simply add their respective percentages:

• If juniors are 40% and seniors are 30%, then the total is 70%.

Percentages present information in an easily digestible format, offering immediate insights into the data distribution.