Question

assume that females have pulse rates that are normally distributed with a mean of $mu = 74.0$ beats per minute and a standard deviation of $sigma = 12.5$ beats per minute. complete parts (a) through (c) below.
the probability is 0.8849.
(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 distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size.
○ b. since the original population has a normal distribution, the distribution of sample means 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 mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size.

Explanation:

The Central Limit Theorem states that if the original population is normally distributed, the sampling distribution of the sample mean \(\bar{x}\) is also normally distributed, regardless of the sample size \(n\). In this case, the population of female pulse - rates is normally distributed. So, for part (b), even if the sample size does not exceed 30, the distribution of sample means is normal.

• Option A is incorrect because part (b) is likely about sample means (not explicitly stated in the problem, but based on the context of using the normal distribution for non - large sample sizes).
• Option C is incorrect. The fact that it is a distribution of sample means is not the sole reason. The original population's normality is crucial.
• Option D is incorrect. The mean of the population (\(\mu = 74\)) has no relation to the application of the normal distribution for sample means in terms of the sample - size condition.

Answer:

B. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.