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?

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.

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%.

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%.

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.