Questions: Assume that females have pulse rates that are normally distributed with a mean of μ=75.0 beats per minute and a standard deviation of σ=12.5 beats per minute. Complete parts (a) through (c) below. a. If 1 adult female is randomly selected, find the probability that her pulse rate is less than 82 beats per minute. The probability is .7123 . (Round to four decimal places as needed.) b. If 16 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 82 beats per minute. The probability is .9875 . (Round to four decimal places as needed.) c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30 ? A. Since the mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size. B. Since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size. C. Since the distribution is of sample means, not individuals, the distribution is a normal distribution for any sample size. D. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.

Assume that females have pulse rates that are normally distributed with a mean of μ=75.0 beats per minute and a standard deviation of σ=12.5 beats per minute. Complete parts (a) through (c) below. a. If 1 adult female is randomly selected, find the probability that her pulse rate is less than 82 beats per minute. The probability is .7123 . (Round to four decimal places as needed.) b. If 16 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 82 beats per minute. The probability is .9875 . (Round to four decimal places as needed.) c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30 ? A. Since the mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size. B. Since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size. C. Since the distribution is of sample means, not individuals, the distribution is a normal distribution for any sample size. D. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.
Transcript text: Assume that females have pulse rates that are normally distributed with a mean of $\mu=75.0$ beats per minute and a standard deviation of $\sigma=12.5$ beats per minute. Complete parts (a) through (c) below. a. If 1 adult female is randomly selected, find the probability that her pulse rate is less than 82 beats per minute. The probability is .7123 . (Round to four decimal places as needed.) b. If 16 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 82 beats per minute. The probability is .9875 . (Round to four decimal places as needed.) c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30 ? A. Since the mean pulse rate exceeds 30 , the distribution of sample means is a normal distribution for any sample size. B. Since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size. C. Since the distribution is of sample means, not individuals, the distribution is a normal distribution for any sample size. D. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.
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Solution

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Solution Steps

Step 1: Calculate the Z-Score for Part (a)

To find the probability that a randomly selected adult female has a pulse rate less than \( 82 \) beats per minute, we first calculate the Z-score using the formula:

\[ z = \frac{X - \mu}{\sigma} = \frac{82 - 75.0}{12.5} = 0.56 \]

Step 2: Calculate the Probability for Part (a)

Next, we use the Z-score to find the probability. The probability that the pulse rate is less than \( 82 \) beats per minute is given by:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.56) - \Phi(-\infty) = 0.7123 \]

Step 3: Calculate the Probability for Part (b)

For part (b), we need to find the probability that the mean pulse rate of \( 16 \) adult females is less than \( 82 \) beats per minute. We calculate the Z-score for the sample mean, where the standard deviation of the sample mean is given by \( \frac{\sigma}{\sqrt{n}} \):

\[ z = \frac{82 - 75.0}{\frac{12.5}{\sqrt{16}}} = \frac{82 - 75.0}{3.125} = 2.24 \]

Now, we find the probability:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(2.24) - \Phi(-\infty) = 0.9875 \]

Step 4: Explanation for Part (c)

The reason we can use the normal distribution in part (b), even though the sample size does not exceed \( 30 \), is:

\[ \text{D. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.} \]

Final Answer

  • Probability for part (a): \( \boxed{0.7123} \)
  • Probability for part (b): \( \boxed{0.9875} \)
  • Explanation for part (c): \( \boxed{\text{D}} \)
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