To find the first and third quartiles (Q1 and Q3) for the class, we can use Z-scores along with the mean and standard deviation provided.
The Z-score formula is given by:
Z = (X - μ) / σ
Where:
- Z is the Z-score
- X is the value we want to convert to a Z-score
- μ is the population mean
- σ is the standard deviation
To find the quartiles, we need to find the corresponding Z-scores for the desired probabilities (P), which in this case are P(Z < 0.25) and P(Z < 0.75).
1. To find the first quartile (Q1):
- We want to find the Z-score that corresponds to P(Z < 0.25).
- Using a Z-table or a calculator, we can find that the Z-score corresponding to P(Z < 0.25) is approximately -0.674.
- Now we can rearrange the Z-score formula to solve for X:
X = Z * σ + μ
- Plugging in the known values:
X = (-0.674) * 3.03 + 21.9
- Calculating the result:
X ≈ 19.74
- So, the first quartile (Q1) for the class is approximately 19.74.
2. To find the third quartile (Q3):
- We want to find the Z-score that corresponds to P(Z < 0.75).
- Using a Z-table or a calculator, we can find that the Z-score corresponding to P(Z < 0.75) is approximately 0.674.
- Again, rearranging the Z-score formula to solve for X:
X = Z * σ + μ
- Plugging in the known values:
X = (0.674) * 3.03 + 21.9
- Calculating the result:
X ≈ 23.06
- So, the third quartile (Q3) for the class is approximately 23.06.
By using the Z-score formula and the provided mean and standard deviation, we were able to calculate the approximate values of Q1 and Q3 for the class ages.