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Part 1
Assume that females have pulse rates that are normally distributed with a mean of mu equals 73.0 beats per minute and a standard deviation of sigma equals 12.5 beats per minute. Complete parts​ (a) through​ (c) below.
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Part 1
a. If 1 adult female is randomly​ selected, find the probability that her pulse rate is less than 80 beats per minute.
The probability is

enter your response here.
​(Round to four decimal places as​ needed.)

We are asked to find the probability that a randomly selected adult female has a pulse rate less than 80 beats per minute, given that the pulse rates of females are normally distributed with a mean (\(\mu\)) of 73 beats per minute and a standard deviation (\(\sigma\)) of 12.5 beats per minute.

This can be solved using the **standard normal distribution** (Z-distribution). The formula to convert a value to a Z-score is:

\[
Z = \frac{X - \mu}{\sigma}
\]

Where:
- \( X \) is the value we are interested in (80 beats per minute),
- \( \mu \) is the mean (73 beats per minute),
- \( \sigma \) is the standard deviation (12.5 beats per minute).

### Step 1: Calculate the Z-score for \( X = 80 \):

\[
Z = \frac{80 - 73}{12.5} = \frac{7}{12.5} = 0.56
\]

### Step 2: Find the probability corresponding to \( Z = 0.56 \).

Using standard normal distribution tables or a calculator, the cumulative probability for a Z-score of 0.56 is approximately **0.7123**.

### Step 3: Conclusion

The probability that a randomly selected adult female has a pulse rate less than 80 beats per minute is approximately **0.7123** (rounded to four decimal places).

Let me know if you'd like more details on the steps, or have any questions. Here are some related questions:

1. What is the probability that a female's pulse rate is greater than 80 beats per minute?
2. What is the probability that a female's pulse rate is between 70 and 80 beats per minute?
3. If 10 females are selected, what is the probability that the mean pulse rate of this group is less than 80 beats per minute?
4. How would the probability change if the standard deviation were 10 beats per minute?
5. What percentage of females have pulse rates between 60 and 85 beats per minute?

**Tip:** When solving problems involving normally distributed data, converting values to Z-scores simplifies the process of finding probabilities.

Assume that females have pulse rates that are normally distributed with a mean of 73.0 beats per minute and a standard deviation of 12.5 beats per minute. If 1 adult female is randomly selected, find the probability that her pulse rate is less than 80 beats per minute. Round to four decimal places.

Calculate the probability that a randomly selected female has a pulse rate less than 80 beats per minute using the normal distribution. This problem covers core statistical concepts like Z-scores and standard normal distribution. Suitable for students in Grades 10-12.

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