Problem 4 - Algebra 2

Question

The table shows the frequency of responses to editorials. Find each probability. $$ \begin{array}{|l|l|l|l|l|l|l|}\hline \text { Number of Responses } & {0} & {1} & {2} & {3} & {4} & {5} & {6 \text { or more }} & {\text { Total }} \\ \hline \text { Number of Editorials } & {20} & {30} & {56} & {38} & {34} & {16} & {6} & {200} \\ \hline\end{array} $$ \(P(\text { at most } 4 \text { responses })\)

Step-by-Step Solution

Answer

The probability of an editorial receiving at most 4 responses is 0.89.

Step 1 - Determine the Number of Editorials with at Most 4 Responses

To determine the number of editorials with at most 4 responses, add together the number of editorials that have 0 to 4 responses. From the table, this is \(20 + 30 + 56 + 38 + 34 = 178\).

Step 2 - Calculate the Total Number of Editorials

The total number of editorials is listed in the table as 200.

Step 3 - Calculate the Probability

To calculate the probability, divide the number of editorials with at most 4 responses by the total number of editorials. This gives \(178 / 200 = 0.89\).

Key Concepts

Frequency Distribution

Frequency distribution is an essential concept when dealing with statistics. It helps us understand how often different values appear in a data set. In the context of the exercise, the frequency distribution is represented in a table showing the number of editorials and their respective responses. By analyzing this table, you can quickly grasp which response rates are more common.

• The table lists the number of occurrences for each response count, from 0 to 6 or more.
• This allows you to see how responses are distributed across different editorials.
• Frequency helps in identifying patterns and making predictions.

Understanding frequency distribution draws a clearer picture of the data, making further calculations like probability easier.

Editorials

Editorials in this exercise represent articles or pieces of content that received a varying number of responses. Each editorial acts as a unit of analysis, providing data about how often readers engage with them.

• Each editorial can have 0 to 6 or more responses as shown in the table.
• The distribution of responses across 200 editorials gives insight into how engaging or popular the editorials are.

Using editorials helps in understanding reader behavior and their response dynamics. Such information is valuable in fields like media analysis and marketing.

Responses

Responses refer to the number of times readers interacted with the editorials. In this activity, they are counted to assess how editorials perform in engaging the audience.

• Responses can range from no engagement (0 responses) to high engagement (6 or more).
• Analyzing responses helps in evaluating the effectiveness of content.

This analysis provides feedback on what type of content attracts reader reactions, helping in tailoring future editorials to meet audience preferences better.

Probability Calculation

Probability calculation is the method used to determine the likelihood of a specific outcome. In our exercise, it refers to finding how likely it is for an editorial to get at most 4 responses.

• First, determine the total number of such editorials, which is the sum of those with responses ranging from 0 to 4.
• In this case, there are 178 editorials with at most 4 responses.
• The total number of editorials is 200.
• Dividing these gives a probability of 0.89, meaning there's an 89% chance an editorial will receive at most 4 responses.

Understanding probability calculations allows for better data-driven decision making, particularly in predicting future outcomes based on historical data.