Mr. McElroy enjoys listening to his MP3 player, which has 2,000 songs in five different genres of music. He took a sample of the number of each type of song it played on two different one-hour sessions when it was set to play a random mix. The results are displayed in the table below. What is a good prediction for the number of rock songs Mr. McElroy has on his MP3 player?
| | Sample 1 | Sample 2 |
| :------ | :------- | :------- |
| Country | 5 | 3 |
| Pop | 4 | 3 |
| Classical | 3 | 5 |
| Jazz | 2 | 4 |
| Rock | 6 | 5 |
A. 225
B. 550
C. 1100
D. 1667
Answer (2)
Calculate the proportion of rock songs in Sample 1: 20 6 .
Calculate the proportion of rock songs in Sample 2: 20 5 .
Average the two proportions: 2 20 6 + 20 5 = 40 11 .
Multiply the average proportion by the total number of songs: 40 11 × 2000 = 550 .
Explanation
Analyze the data First, let's analyze the data. We have two samples of songs played from Mr. McElroy's MP3 player. We need to estimate the number of rock songs he has based on these samples.
Calculate proportion in Sample 1 Next, we calculate the proportion of rock songs in each sample. In Sample 1, there are 6 rock songs out of a total of 5 (Country) + 4 (Pop) + 3 (Classical) + 2 (Jazz) + 6 (Rock) = 20 songs. So, the proportion of rock songs in Sample 1 is 20 6 .
Calculate proportion in Sample 2 In Sample 2, there are 5 rock songs out of a total of 3 (Country) + 3 (Pop) + 5 (Classical) + 4 (Jazz) + 5 (Rock) = 20 songs. So, the proportion of rock songs in Sample 2 is 20 5 .
Average the proportions Now, we average the proportions from the two samples to get an overall estimated proportion of rock songs: 2 20 6 + 20 5 = 2 20 11 = 40 11
Predict the number of rock songs Finally, we multiply the overall estimated proportion by the total number of songs (2000) to predict the number of rock songs: 40 11 × 2000 = 11 × 50 = 550
State the final answer Therefore, a good prediction for the number of rock songs Mr. McElroy has on his MP3 player is 550.
Examples
Imagine you're trying to estimate how many students in your school like a certain type of music. You survey two different groups of students and find the proportion of students who like that music in each group. Averaging these proportions and multiplying by the total number of students gives you an estimate of how many students in the whole school like that type of music. This is similar to how we estimated the number of rock songs on Mr. McElroy's MP3 player.
Based on the analysis of two song samples, the estimated proportion of rock songs is calculated as 40 11 . Multiplying this proportion by the total number of songs (2000) gives us an estimate of 550 rock songs. Thus, the best prediction is 550 .
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