Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

After the private screening of a new television pilot, audience members were asked to rate the new show on a scale of 1 to 10 ( 10 being the highest rating). From a group of 140 people, the following responses were obtained: $$\begin{array}{lllllllllll}\hline \text { Rating } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text { Frequency of } & & & & & & & & & & \\\\\text { Occurrence } & 1 & 4 & 3 & 11 & 23 & 21 & 28 & 29 & 16 & 4 \\\\\hline\end{array}$$ Let the random variable \(X\) denote the rating given to the show by a randomly chosen audience member. Find the probability distribution associated with these data.

Step-by-Step Solution

Answer

The probability distribution of the random variable \(X\) representing the rating given by a randomly chosen audience member is: $$ \begin{array}{lllllllllll} \hline \text { X } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline P(X) & \frac{1}{140} & \frac{4}{140} & \frac{3}{140} & \frac{11}{140} & \frac{23}{140} & \frac{21}{140} & \frac{28}{140} & \frac{29}{140} & \frac{16}{140} & \frac{4}{140} \\ \hline \end{array} $$

Step 1: Compute the probability of each rating

We are given the frequency of occurrence for each rating: $$ \begin{array}{lllllllllll} \hline \text { Rating } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text { Frequency of } & & & & & & & & & & \\ \text { Occurrence } & 1 & 4 & 3 & 11 & 23 & 21 & 28 & 29 & 16 & 4 \\ \hline \end{array} $$ There are a total of 140 people. To compute the probability of each rating, we will divide the frequency of occurrence for each rating by 140: $$ P(X = 1) = \frac{1}{140}, P(X = 2) = \frac{4}{140}, P(X = 3) = \frac{3}{140}, \ldots, P(X = 10) = \frac{4}{140} $$

Step 2: Represent the probability distribution

Now that we have the probabilities for each rating, let's represent the probability distribution associated with these data. The probability distribution of the random variable \(X\) can be written as: $$ \begin{array}{lllllllllll} \hline \text { X } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline P(X) & \frac{1}{140} & \frac{4}{140} & \frac{3}{140} & \frac{11}{140} & \frac{23}{140} & \frac{21}{140} & \frac{28}{140} & \frac{29}{140} & \frac{16}{140} & \frac{4}{140} \\ \hline \end{array} $$ This represents the probability distribution of the random variable \(X\) denoting the rating given by a randomly chosen audience member.

Key Concepts

Random Variable

A random variable is an integral part of probability and statistics. It represents outcomes of a random phenomenon. In this context, the random variable X is defined as the rating given by an audience member to a television pilot.

Random variables can take on various values depending upon the situation. Here, the values range from 1 to 10, signifying different ratings possible by an audience member. Each value of the random variable is associated with the probability of that value occurring.

Understanding random variables is crucial because they allow us to analyze and interpret outcomes of experiments that have an element of randomness. By defining a random variable, we can use statistical methods to calculate probabilities, averages, and other characteristics of the outcomes.

Frequency Distribution

Frequency distribution is a way to organize data to show the number of times each value occurs. In the exercise, the frequency distribution presents how often each rating was given by the audience. This allows us to understand which ratings were more popular.

The data collected might look like this:

• Rating 1 was given by 1 person
• Rating 2 was given by 4 people
• Rating 3 was given by 3 people
• ... and so on up to Rating 10 given by 4 people

Breaking down data in this manner enables us to quickly view and interpret the information, making it easier to identify trends, such as most common ratings, in the audience's response to the pilot.

Probability Calculation

Probability calculation helps us determine the likelihood of certain outcomes. For each possible rating in this exercise, we calculate the probability by dividing the frequency of that rating by the total number of ratings.

The formula used is: \(P(X = x) = \frac{\text{Frequency of } x}{\text{Total Frequency}}\)
For instance, the probability of a rating of 8 would be computed as: \(P(X = 8) = \frac{29}{140}\)

Understanding these probabilities allows us to construct the probability distribution, a complete list of each rating and the associated probability. This list helps us see how likely an audience member is to give a certain rating, providing valuable insights into audience preferences and general feedback for the show.