Question
The weight of male babies less than 2 months old in the United States is normally distributed with mean 12.4 and standard deviation 5.2. ) What proportion of babies weigh more than 14 pounds?
Answers
GPT-4o mini
To determine the proportion of male babies less than 2 months old who weigh more than 14 pounds, we can use the properties of the normal distribution.
1. **Given Information:**
- Mean (\(\mu\)) = 12.4 pounds
- Standard Deviation (\(\sigma\)) = 5.2 pounds
- We want to find the proportion of babies with weights greater than 14 pounds.
2. **Calculate the Z-score:**
The Z-score can be calculated using the formula:
\[
Z = \frac{(X - \mu)}{\sigma}
\]
where:
- \(X\) is the value we are interested in (14 pounds)
- \(\mu\) is the mean (12.4)
- \(\sigma\) is the standard deviation (5.2)
Now let's calculate the Z-score:
\[
Z = \frac{(14 - 12.4)}{5.2} = \frac{1.6}{5.2} \approx 0.3077
\]
3. **Find the Proportion:**
To find the proportion of babies weighing more than 14 pounds, we need to look up the Z-score in the standard normal distribution table, or use a calculator or software that provides the cumulative distribution function (CDF) for the normal distribution.
The CDF will give us the area to the left of the Z-score.
- Looking up \(Z = 0.3077\), we find that the area to the left of this Z-score is approximately 0.6217 (this value may vary slightly depending on the table or software used).
4. **Calculate the Proportion Above 14 Pounds:**
Since we are interested in the proportion that is greater than 14 pounds, we take the complement of the CDF:
\[
P(X > 14) = 1 - P(X \leq 14)
\]
\[
P(X > 14) = 1 - 0.6217 = 0.3783
\]
**Conclusion:** Approximately 37.83% of male babies less than 2 months old weigh more than 14 pounds.
1. **Given Information:**
- Mean (\(\mu\)) = 12.4 pounds
- Standard Deviation (\(\sigma\)) = 5.2 pounds
- We want to find the proportion of babies with weights greater than 14 pounds.
2. **Calculate the Z-score:**
The Z-score can be calculated using the formula:
\[
Z = \frac{(X - \mu)}{\sigma}
\]
where:
- \(X\) is the value we are interested in (14 pounds)
- \(\mu\) is the mean (12.4)
- \(\sigma\) is the standard deviation (5.2)
Now let's calculate the Z-score:
\[
Z = \frac{(14 - 12.4)}{5.2} = \frac{1.6}{5.2} \approx 0.3077
\]
3. **Find the Proportion:**
To find the proportion of babies weighing more than 14 pounds, we need to look up the Z-score in the standard normal distribution table, or use a calculator or software that provides the cumulative distribution function (CDF) for the normal distribution.
The CDF will give us the area to the left of the Z-score.
- Looking up \(Z = 0.3077\), we find that the area to the left of this Z-score is approximately 0.6217 (this value may vary slightly depending on the table or software used).
4. **Calculate the Proportion Above 14 Pounds:**
Since we are interested in the proportion that is greater than 14 pounds, we take the complement of the CDF:
\[
P(X > 14) = 1 - P(X \leq 14)
\]
\[
P(X > 14) = 1 - 0.6217 = 0.3783
\]
**Conclusion:** Approximately 37.83% of male babies less than 2 months old weigh more than 14 pounds.
(b) What proportion of babies weigh less than 15 pounds
GPT-4o mini
To find the proportion of male babies less than 2 months old who weigh less than 15 pounds, we can again use the properties of the normal distribution.
1. **Given Information:**
- Mean (\(\mu\)) = 12.4 pounds
- Standard Deviation (\(\sigma\)) = 5.2 pounds
- We're interested in the weight \(X = 15\) pounds.
2. **Calculate the Z-score:**
The Z-score is calculated using the formula:
\[
Z = \frac{(X - \mu)}{\sigma}
\]
Substituting the values:
\[
Z = \frac{(15 - 12.4)}{5.2} = \frac{2.6}{5.2} \approx 0.5
\]
3. **Find the Proportion:**
Next, we look up the Z-score in the standard normal distribution table or use a calculator that provides the cumulative distribution function (CDF) for the normal distribution.
- Finding the CDF for \(Z = 0.5\), we can determine that the area to the left (the proportion of babies weighing less than 15 pounds) is approximately 0.6915.
4. **Conclusion:**
Therefore, the proportion of male babies less than 2 months old that weigh less than 15 pounds is approximately **69.15%**.
1. **Given Information:**
- Mean (\(\mu\)) = 12.4 pounds
- Standard Deviation (\(\sigma\)) = 5.2 pounds
- We're interested in the weight \(X = 15\) pounds.
2. **Calculate the Z-score:**
The Z-score is calculated using the formula:
\[
Z = \frac{(X - \mu)}{\sigma}
\]
Substituting the values:
\[
Z = \frac{(15 - 12.4)}{5.2} = \frac{2.6}{5.2} \approx 0.5
\]
3. **Find the Proportion:**
Next, we look up the Z-score in the standard normal distribution table or use a calculator that provides the cumulative distribution function (CDF) for the normal distribution.
- Finding the CDF for \(Z = 0.5\), we can determine that the area to the left (the proportion of babies weighing less than 15 pounds) is approximately 0.6915.
4. **Conclusion:**
Therefore, the proportion of male babies less than 2 months old that weigh less than 15 pounds is approximately **69.15%**.
(c) What proportion of babies weigh between 10.3 and 14.9 pounds ?
GPT-4o mini
To find the proportion of male babies less than 2 months old who weigh between 10.3 and 14.9 pounds, we will calculate the Z-scores for both weight values and then find the corresponding probabilities using the standard normal distribution.
### Given Information:
- Mean (\(\mu\)) = 12.4 pounds
- Standard Deviation (\(\sigma\)) = 5.2 pounds
### Step 1: Calculate the Z-scores
1. **For 10.3 pounds:**
\[
Z_{10.3} = \frac{(10.3 - 12.4)}{5.2} = \frac{-2.1}{5.2} \approx -0.4038
\]
2. **For 14.9 pounds:**
\[
Z_{14.9} = \frac{(14.9 - 12.4)}{5.2} = \frac{2.5}{5.2} \approx 0.4808
\]
### Step 2: Find the Proportions
Next, we need to look up the Z-scores in the standard normal distribution table or use a calculator.
1. **Looking up \(Z_{10.3} \approx -0.4038\)**:
- The cumulative probability \(P(Z < -0.4038)\) is approximately **0.3424**.
2. **Looking up \(Z_{14.9} \approx 0.4808\)**:
- The cumulative probability \(P(Z < 0.4808)\) is approximately **0.6844**.
### Step 3: Calculate the Proportion Between the Two Values
To find the proportion of babies weighing between 10.3 and 14.9 pounds, we can subtract the two probabilities:
\[
P(10.3 < X < 14.9) = P(Z < 0.4808) - P(Z < -0.4038)
\]
\[
P(10.3 < X < 14.9) = 0.6844 - 0.3424 = 0.3420
\]
### Conclusion
Therefore, the proportion of male babies less than 2 months old who weigh between 10.3 and 14.9 pounds is approximately **34.20%**.
### Given Information:
- Mean (\(\mu\)) = 12.4 pounds
- Standard Deviation (\(\sigma\)) = 5.2 pounds
### Step 1: Calculate the Z-scores
1. **For 10.3 pounds:**
\[
Z_{10.3} = \frac{(10.3 - 12.4)}{5.2} = \frac{-2.1}{5.2} \approx -0.4038
\]
2. **For 14.9 pounds:**
\[
Z_{14.9} = \frac{(14.9 - 12.4)}{5.2} = \frac{2.5}{5.2} \approx 0.4808
\]
### Step 2: Find the Proportions
Next, we need to look up the Z-scores in the standard normal distribution table or use a calculator.
1. **Looking up \(Z_{10.3} \approx -0.4038\)**:
- The cumulative probability \(P(Z < -0.4038)\) is approximately **0.3424**.
2. **Looking up \(Z_{14.9} \approx 0.4808\)**:
- The cumulative probability \(P(Z < 0.4808)\) is approximately **0.6844**.
### Step 3: Calculate the Proportion Between the Two Values
To find the proportion of babies weighing between 10.3 and 14.9 pounds, we can subtract the two probabilities:
\[
P(10.3 < X < 14.9) = P(Z < 0.4808) - P(Z < -0.4038)
\]
\[
P(10.3 < X < 14.9) = 0.6844 - 0.3424 = 0.3420
\]
### Conclusion
Therefore, the proportion of male babies less than 2 months old who weigh between 10.3 and 14.9 pounds is approximately **34.20%**.
(d) Is it unusual for a baby to weigh more than 17 pounds ? why or why not
GPT-4o mini
To determine whether it is unusual for a baby less than 2 months old to weigh more than 17 pounds, we can calculate the Z-score for 17 pounds and then find the corresponding cumulative probability. In general, if a measurement falls beyond 2 standard deviations from the mean, it is often considered unusual (or an outlier).
### Given Information:
- Mean (\(\mu\)) = 12.4 pounds
- Standard Deviation (\(\sigma\)) = 5.2 pounds
- Weight of interest (\(X = 17\) pounds)
### Step 1: Calculate the Z-score
Using the formula for Z-score:
\[
Z = \frac{(X - \mu)}{\sigma}
\]
Substituting the values:
\[
Z = \frac{(17 - 12.4)}{5.2} = \frac{4.6}{5.2} \approx 0.8846
\]
### Step 2: Find the Cumulative Probability
Next, we can look up the Z-score in the standard normal distribution table or use a calculator to find the cumulative probability.
- The cumulative probability for \(Z \approx 0.8846\) is approximately **0.8120**. This means that about 81.20% of babies weigh less than 17 pounds.
### Step 3: Find the Proportion of Babies Weighing More Than 17 Pounds
To find the proportion of babies that weigh more than 17 pounds, we can subtract this cumulative probability from 1:
\[
P(X > 17) = 1 - P(X \leq 17)
\]
\[
P(X > 17) = 1 - 0.8120 = 0.1880
\]
### Step 4: Interpret the Result
A proportion of 0.1880 (or 18.80%) means that around 18.80% of babies weigh more than 17 pounds.
### Conclusion
- **Is it unusual?**
- While it may not be common for a baby to weigh more than 17 pounds (as approximately 18.80% of babies do), it does not fall beyond the typical threshold of 2 standard deviations from the mean. A Z-score of approximately 0.8846 indicates that 17 pounds is actually within the range of typical weights for babies less than 2 months old. Thus, while it is on the higher side, it is not statistically unusual for a baby to weigh more than 17 pounds.
### Given Information:
- Mean (\(\mu\)) = 12.4 pounds
- Standard Deviation (\(\sigma\)) = 5.2 pounds
- Weight of interest (\(X = 17\) pounds)
### Step 1: Calculate the Z-score
Using the formula for Z-score:
\[
Z = \frac{(X - \mu)}{\sigma}
\]
Substituting the values:
\[
Z = \frac{(17 - 12.4)}{5.2} = \frac{4.6}{5.2} \approx 0.8846
\]
### Step 2: Find the Cumulative Probability
Next, we can look up the Z-score in the standard normal distribution table or use a calculator to find the cumulative probability.
- The cumulative probability for \(Z \approx 0.8846\) is approximately **0.8120**. This means that about 81.20% of babies weigh less than 17 pounds.
### Step 3: Find the Proportion of Babies Weighing More Than 17 Pounds
To find the proportion of babies that weigh more than 17 pounds, we can subtract this cumulative probability from 1:
\[
P(X > 17) = 1 - P(X \leq 17)
\]
\[
P(X > 17) = 1 - 0.8120 = 0.1880
\]
### Step 4: Interpret the Result
A proportion of 0.1880 (or 18.80%) means that around 18.80% of babies weigh more than 17 pounds.
### Conclusion
- **Is it unusual?**
- While it may not be common for a baby to weigh more than 17 pounds (as approximately 18.80% of babies do), it does not fall beyond the typical threshold of 2 standard deviations from the mean. A Z-score of approximately 0.8846 indicates that 17 pounds is actually within the range of typical weights for babies less than 2 months old. Thus, while it is on the higher side, it is not statistically unusual for a baby to weigh more than 17 pounds.