Answer :
To analyze Josiah's prediction, we break down the information provided and Josiah's methodology clearly and step-by-step.
### Step-by-Step Analysis
1. Identify the counts of each genre from the samples:
- Sample 1:
- R&B: 5 songs
- Pop: 4 songs
- Classical: 3 songs
- Jazz: 2 songs
- Rock: 6 songs
- Sample 2:
- R&B: 3 songs
- Pop: 3 songs
- Classical: 5 songs
- Jazz: 4 songs
- Rock: 4 songs
2. Calculate the total number of songs in each sample:
- Total Sample 1: [tex]\(5 + 4 + 3 + 2 + 6 = 20\)[/tex] songs
- Total Sample 2: [tex]\(3 + 3 + 5 + 4 + 4 = 19\)[/tex] songs
3. Count the number of Rock songs in each sample:
- Rock songs in Sample 1: 6
- Rock songs in Sample 2: 4
4. Calculate the total number of songs across both samples:
- Total songs in samples: [tex]\(20 + 19 = 39\)[/tex]
5. Calculate the total number of Rock songs across both samples:
- Total Rock songs in samples: [tex]\(6 + 4 = 10\)[/tex]
6. Calculate the proportion of Rock songs in the samples:
- Proportion of Rock songs: [tex]\(\frac{10}{39}\)[/tex]
7. Use the proportion to predict the number of Rock songs on the MP3 player:
- Total songs on MP3 player: 1500
- Predicted number of Rock songs: [tex]\(\left(\frac{10}{39}\right) \times 1500 \approx 384.615\)[/tex]
Josiah predicted the number of Rock songs to be 300 songs, but the proportionate calculation predicts about 384.615 songs, which is a significant difference.
### Analyzing Josiah's Statements
Considering Josiah’s work:
[tex]\[ \frac{10}{20} = \frac{x}{1500} \][/tex]
This formulation is incorrect due to:
1. Incorrect Denominator: He incorrectly used 20 (perhaps misinterpreting Sample 1's total) instead of the combined total from both samples (39).
2. Misalignment with Proportional Calculation: The actual combined total count (both samples) calculates about 384.615 Rock Songs, not the 300 songs Josiah predicted.
Now, we analyze statements based on Josiah’s solution:
1. Josiah's prediction method and the numbers he used are inconsistent.
2. The correct fraction to use from samples would involve the total counts (39).
3. With the correct calculation, the predicted Rock songs on a 1500-song MP3 player should be around 384.615.
### Conclusion
Without referring to what might be Josiah’s logic errors, the verified and consistent results portray the correct prediction and hence:
- Josiah’s calculation of the fraction is flawed.
- Using the correct combined counts significantly alters the prediction.
Thus, these statements about Josiah's work can be identified as true:
1. Josiah predicted the number of Rock songs using an incorrect fraction.
2. Calculating with the correct sample proportions shows a prediction of around 384.615 Rock songs.
3. Josiah's value of 300 Rock songs is inconsistent with calculated proportional prediction.
### Step-by-Step Analysis
1. Identify the counts of each genre from the samples:
- Sample 1:
- R&B: 5 songs
- Pop: 4 songs
- Classical: 3 songs
- Jazz: 2 songs
- Rock: 6 songs
- Sample 2:
- R&B: 3 songs
- Pop: 3 songs
- Classical: 5 songs
- Jazz: 4 songs
- Rock: 4 songs
2. Calculate the total number of songs in each sample:
- Total Sample 1: [tex]\(5 + 4 + 3 + 2 + 6 = 20\)[/tex] songs
- Total Sample 2: [tex]\(3 + 3 + 5 + 4 + 4 = 19\)[/tex] songs
3. Count the number of Rock songs in each sample:
- Rock songs in Sample 1: 6
- Rock songs in Sample 2: 4
4. Calculate the total number of songs across both samples:
- Total songs in samples: [tex]\(20 + 19 = 39\)[/tex]
5. Calculate the total number of Rock songs across both samples:
- Total Rock songs in samples: [tex]\(6 + 4 = 10\)[/tex]
6. Calculate the proportion of Rock songs in the samples:
- Proportion of Rock songs: [tex]\(\frac{10}{39}\)[/tex]
7. Use the proportion to predict the number of Rock songs on the MP3 player:
- Total songs on MP3 player: 1500
- Predicted number of Rock songs: [tex]\(\left(\frac{10}{39}\right) \times 1500 \approx 384.615\)[/tex]
Josiah predicted the number of Rock songs to be 300 songs, but the proportionate calculation predicts about 384.615 songs, which is a significant difference.
### Analyzing Josiah's Statements
Considering Josiah’s work:
[tex]\[ \frac{10}{20} = \frac{x}{1500} \][/tex]
This formulation is incorrect due to:
1. Incorrect Denominator: He incorrectly used 20 (perhaps misinterpreting Sample 1's total) instead of the combined total from both samples (39).
2. Misalignment with Proportional Calculation: The actual combined total count (both samples) calculates about 384.615 Rock Songs, not the 300 songs Josiah predicted.
Now, we analyze statements based on Josiah’s solution:
1. Josiah's prediction method and the numbers he used are inconsistent.
2. The correct fraction to use from samples would involve the total counts (39).
3. With the correct calculation, the predicted Rock songs on a 1500-song MP3 player should be around 384.615.
### Conclusion
Without referring to what might be Josiah’s logic errors, the verified and consistent results portray the correct prediction and hence:
- Josiah’s calculation of the fraction is flawed.
- Using the correct combined counts significantly alters the prediction.
Thus, these statements about Josiah's work can be identified as true:
1. Josiah predicted the number of Rock songs using an incorrect fraction.
2. Calculating with the correct sample proportions shows a prediction of around 384.615 Rock songs.
3. Josiah's value of 300 Rock songs is inconsistent with calculated proportional prediction.
