Bot GPT 4-o mini To find the requested percentiles and quartile for the normally distributed weights of male babies less than 2 months old, we first need to utilize the properties of the normal distribution.
Given:
- Mean (μ) = 11.9 pounds
- Standard Deviation (σ) = 3.5 pounds
We'll use the z-score formula and the standard normal distribution to find the percentiles. The z-score can be calculated with the following formula:
\[ z = \frac{X - \mu}{\sigma} \]
Where:
- \(X\) is the value in the original distribution
- \(μ\) is the mean
- \(σ\) is the standard deviation
To reverse this, we can solve for \(X\):
\[ X = μ + zσ \]
(a) Find the 83rd percentile of the baby weights.
- Look up the z-score corresponding to the 83rd percentile in the standard normal distribution table (or use a calculator). The z-score for the 83rd percentile is approximately 0.954.
- Now, use the formula to find \(X\):
\[ X = 11.9 + (0.954 \times 3.5) \]
Calculating:
\[ X \approx 11.9 + 3.339 = 15.239 \]
So, the 83rd percentile of the baby weights is approximately 15.24 pounds.
(b) Find the 12th percentile of the baby weights.
- Look up the z-score corresponding to the 12th percentile. The z-score for the 12th percentile is approximately -1.175.
- Now, use the formula to find \(X\):
\[ X = 11.9 + (-1.175 \times 3.5) \]
Calculating:
\[ X \approx 11.9 - 4.1125 = 7.7875 \]
So, the 12th percentile of the baby weights is approximately 7.79 pounds.
(c) Find the first quartile of the baby weights.
The first quartile (Q1) corresponds to the 25th percentile.
- Look up the z-score corresponding to the 25th percentile. The z-score for the 25th percentile is approximately -0.675.
- Now, use the formula to find \(X\):
\[ X = 11.9 + (-0.675 \times 3.5) \]
Calculating:
\[ X \approx 11.9 - 2.3625 = 9.5375 \]
So, the first quartile (Q1) of the baby weights is approximately 9.54 pounds.
Final Results:
- (a) 83rd Percentile: 15.24 pounds
- (b) 12th Percentile: 7.79 pounds
- (c) First Quartile (Q1): 9.54 pounds
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