Part 1a:

We are given that the pulse rates of females are normally distributed with:

• Mean, $\mu = 75.0$ beats per minute,
• Standard deviation, $\sigma = 12.5$ beats per minute.

We are asked to find the probability that a randomly selected adult female has a pulse rate less than 81 beats per minute.

To find this probability, we will use the Z-score formula:

$Z = \frac{X - \mu}{\sigma}$

Where:

• $X$ is the value we're interested in (81 beats per minute),
• $\mu = 75.0$ is the mean,
• $\sigma = 12.5$ is the standard deviation.

Substituting the given values:

$Z = \frac{81 - 75}{12.5} = \frac{6}{12.5} = 0.48$

Now, we need to find the probability corresponding to this Z-score. Using standard normal distribution tables or a calculator, the probability corresponding to $Z = 0.48$ is:

$P(Z < 0.48) \approx 0.6844$

So, the probability that a randomly selected female has a pulse rate less than 81 beats per minute is approximately:

$P(X < 81) \approx 0.6844$

Part 1b:

Now, we're asked to find the probability that the mean pulse rate of 25 randomly selected females is less than 81 beats per minute.

For this, we need to adjust the standard deviation to account for the sample size. The standard error of the sample mean is given by:

$SE = \frac{\sigma}{\sqrt{n}}$

Where:

• $n = 25$ is the sample size,
• $\sigma = 12.5$ is the standard deviation.

So, the standard error is:

$SE = \frac{12.5}{\sqrt{25}} = \frac{12.5}{5} = 2.5$

Now, we will compute the Z-score for the sample mean of 81 beats per minute:

$Z = \frac{X_{\text{mean}} - \mu}{SE} = \frac{81 - 75}{2.5} = \frac{6}{2.5} = 2.4$

Using the Z-table or a calculator, the probability corresponding to $Z = 2.4$ is:

$P(Z < 2.4) \approx 0.9918$

So, the probability that the mean pulse rate of 25 randomly selected females is less than 81 beats per minute is approximately:

$P(\overline{X} < 81) \approx 0.9918$

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Final Answers:

• Part 1a: The probability that a randomly selected female has a pulse rate less than 81 beats per minute is 0.6844.
• Part 1b: The probability that the mean pulse rate of 25 randomly selected females is less than 81 beats per minute is 0.9918.

Would you like further clarification on any step? Here are some related questions that might help:

1. How do you calculate the Z-score for sample means?
2. What happens to the probability when the sample size increases?
3. How do you interpret the Z-table?
4. What other distributions can be used for different types of data?
5. How does the standard deviation of a population affect the Z-score?

Tip: Always remember that for sample means, the standard error is key to adjusting the variability, and it decreases as the sample size increases, making the distribution of the sample mean more concentrated around the population mean.