Practise validating binomial models, calculating their parameters and probabilities, and interpreting success counts or unusual tails in the original context.
Ms. Fey is a manager at a restaurant. To improve the dining experience for her customers, she uses a digital music service to create a playlist of songs that will be played in the restaurant.
The playlist contains 1,000 songs and consists of four different types of music in the following quantities: 200 country songs, 400 pop songs, 100 rock songs, and 300 jazz songs. The digital music service will select songs at random from the playlist to be played in the restaurant. Any song can be replayed at any time.
| Scoring for Question 3 | Score |
|---|---|
| Complete Response | 4 |
| Three parts essentially correct | |
| Substantial Response | 3 |
| Two parts essentially correct and one part partially correct | |
| Developing Response | 2 |
| Two parts essentially correct and no part partially correct OR | |
| One part essentially correct and one or two parts partially correct OR Three parts partially correct | |
| Minimal Response | 1 |
| One part essentially correct and no parts partially correct OR | |
| No part essentially correct and two parts partially correct |
In every one-hour period, 20 songs will be played at random and any song can be replayed at any time. Ms. Fey is interested in how many rock songs will be played in a typical one-hour period.
Define the random variable of interest to Ms. Fey, and state how the random variable is distributed.
Model Solution
Let X be the number of rock songs among the 20 songs played in one hour. Since songs are selected independently with replacement and p=100/1000=0.10, X has a binomial distribution with n=20 and p=0.10.
Scoring components
1. Defines the random variable as the number of rock songs in one hour.
2. Identifies a binomial distribution.
3. States n=20 and p=0.10.
What is the expected value for the random variable in part B (i)? Show your work.
Model Solution
E(X)=np=20(0.10)=2 songs.
Scoring component
4. Correctly calculates the expected value and shows supporting work. The parameters may also be stated here.
Recall that in every one-hour period, 20 songs will be played at random and any song can be replayed at any time.
Determine the probability that 4 or more rock songs in a particular one-hour period will be played. Show your work.
Model Solution
P(X≥4)=1−P(X≤3)=1−x=0∑3(x20)(0.10)x(0.90)20−x=1−0.867=0.133.
Scoring components
1. Gives the correct probability.
2. Shows work supporting the probability.
Suppose 4 rock songs are played during a particular one-hour period. Does this provide strong evidence that the song selection process was not truly random? Justify your answer without performing an inference procedure.
Model Solution
No. The probability of four or more rock songs is 0.133, which is high enough to be reasonably attributed to chance alone and is not small enough to provide strong evidence that the selection process was not truly random.
Scoring components
3. Indicates there is not a strong reason to believe the process was not random.
4. Correctly links the probability to that decision.