Suppose the following data represent the ratings (on a scale from 1 to 5) for a certain smart phone game, with 1 representing a poor rating. Complete parts (a) through (d) below.
\begin{tabular}{c|c}
Stars & Frequency \\
\hline 1 & 2982 \\
2 & 2783 \\
3 & 3852 \\
4 & 4078 \\
5 & 11,551
\end{tabular}
(a) Construct a discrete probability distribution for the random variable [tex]$x$[/tex].
\begin{tabular}{c|c}
Stars ([tex]$x$[/tex]) & [tex]$P(x)$[/tex] \\
\hline 1 & $\square$ \\
2 & $\square$ \\
3 & $\square$ \\
4 & $\square$ \\
5 & $\square$
\end{tabular}
(Round to three decimal places as needed.)
Answer (1)
Calculate the total number of ratings: 2982 + 2783 + 3852 + 4078 + 11551 = 25246 .
Calculate the probability for each star rating by dividing its frequency by the total number of ratings.
Round each probability to three decimal places.
The discrete probability distribution is: P ( 1 ) = 0.118 , P ( 2 ) = 0.110 , P ( 3 ) = 0.153 , P ( 4 ) = 0.162 , P ( 5 ) = 0.458 .
Explanation
Analyze the problem We are given the frequency of star ratings for a smartphone game and asked to construct a discrete probability distribution. This involves calculating the probability of each star rating based on the provided frequencies.
Calculate the total number of ratings First, we need to calculate the total number of ratings. This is done by summing the frequencies of all star ratings:
T o t a l = 2982 + 2783 + 3852 + 4078 + 11551
Total number of ratings The sum of the frequencies is:
T o t a l = 25246
Calculate probabilities for each star rating Next, we calculate the probability for each star rating by dividing its frequency by the total number of ratings. We will round each probability to three decimal places.
P ( 1 ) = 25246 2982 P ( 2 ) = 25246 2783 P ( 3 ) = 25246 3852 P ( 4 ) = 25246 4078 P ( 5 ) = 25246 11551
Calculate the probabilities Now, let's calculate the probabilities:
P ( 1 ) = 25246 2982 ≈ 0.118 P ( 2 ) = 25246 2783 ≈ 0.110 P ( 3 ) = 25246 3852 ≈ 0.153 P ( 4 ) = 25246 4078 ≈ 0.162 P ( 5 ) = 25246 11551 ≈ 0.458
Discrete probability distribution Finally, we present the discrete probability distribution in a table:
Stars ( x )
P ( x )
1
0.118
2
0.110
3
0.153
4
0.162
5
0.458
Examples
Understanding probability distributions is crucial in many real-world scenarios. For instance, in marketing, companies use them to analyze customer satisfaction based on survey ratings. If a product consistently receives high ratings (4 or 5 stars), it indicates customer satisfaction. Conversely, a high frequency of low ratings (1 or 2 stars) suggests areas needing improvement. This analysis helps businesses make informed decisions about product development, marketing strategies, and customer service enhancements.
