Question
The weight of male babies less than 2 months old in the United States is normally distributed with mean 11.9 pounds and standard deviation 3.5 pounds.
(a) Find the 83rd percentile of the baby weights.
(b) Find the 12th percentile of the baby weights.
(c) Find the first quartile of the baby weights.
(a) Find the 83rd percentile of the baby weights.
(b) Find the 12th percentile of the baby weights.
(c) Find the first quartile of the baby weights.
Answers
GPT-4o 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.
1. 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**.
2. 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.
1. Look up the z-score corresponding to the 12th percentile. The z-score for the 12th percentile is approximately **-1.175**.
2. 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.
1. Look up the z-score corresponding to the 25th percentile. The z-score for the 25th percentile is approximately **-0.675**.
2. 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**
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.
1. 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**.
2. 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.
1. Look up the z-score corresponding to the 12th percentile. The z-score for the 12th percentile is approximately **-1.175**.
2. 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.
1. Look up the z-score corresponding to the 25th percentile. The z-score for the 25th percentile is approximately **-0.675**.
2. 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**