Some sources report that the weights of​ full-term newborn babies in a certain town have a mean of 9 pounds and a standard deviation of 1.2 pounds and are Normally distributed.
a. What is the probability that one newborn baby will have a weight within 1.2 pounds of the meanlong dashthat ​is, between 7.8 and 10.2 ​pounds, or within one standard deviation of the​ mean?

To solve this problem, we need to calculate the probability that a newborn baby’s weight falls within 1 standard deviation of the mean in a normally distributed population.

We know:
- The mean \(\mu = 9\) pounds,
- The standard deviation \(\sigma = 1.2\) pounds,
- The weight range between 7.8 pounds and 10.2 pounds is equivalent to being within 1 standard deviation of the mean (since \(9 - 1.2 = 7.8\) and \(9 + 1.2 = 10.2\)).

### Step 1: Standardize the values
To find the probability, we need to convert the given range into **z-scores** using the formula:

\[
z = \frac{x - \mu}{\sigma}
\]

For the lower bound (\(x = 7.8\)):
\[
z_{\text{lower}} = \frac{7.8 - 9}{1.2} = \frac{-1.2}{1.2} = -1
\]

For the upper bound (\(x = 10.2\)):
\[
z_{\text{upper}} = \frac{10.2 - 9}{1.2} = \frac{1.2}{1.2} = 1
\]

### Step 2: Find the probability corresponding to these z-scores
For a standard normal distribution, the probability of a z-score falling between \(-1\) and \(1\) is approximately \(0.6826\). This is a well-known result for the normal distribution.

### Conclusion:
The probability that a newborn baby’s weight is between 7.8 pounds and 10.2 pounds (i.e., within 1 standard deviation of the mean) is **0.6826** or **68.26%**.

Would you like further details on this process or additional questions to explore?

Here are five related questions for practice:
1. What is the probability of a newborn weighing more than 10.2 pounds?
2. What is the probability of a newborn weighing less than 7.8 pounds?
3. How would the probability change if the standard deviation were smaller?
4. What percentage of newborns would weigh between 6.6 pounds and 11.4 pounds (within 2 standard deviations of the mean)?
5. How would you calculate the probability for a newborn weighing more than 12 pounds?

**Tip:** When working with normal distributions, always standardize values to z-scores to use standard normal probability tables effectively.

Some sources report that the weights of full-term newborn babies in a certain town have a mean of 9 pounds and a standard deviation of 1.2 pounds and are Normally distributed. What is the probability that one newborn baby will have a weight within 1.2 pounds of the mean—that is, between 7.8 and 10.2 pounds, or within one standard deviation of the mean?

Calculate the probability that a newborn baby’s weight falls within 1 standard deviation of the mean using a normal distribution with a mean of 9 pounds and a standard deviation of 1.2 pounds. This problem explains key concepts such as z-scores and the empirical rule, suitable for students in Grades 10-12.

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