Question

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. Cormplete parts (a) through (c) below a. If 1 adult female is randomly selected, find the probability that her pulse rate is less than 79 beats per minute. The probability is (Round to four decimal places as needed.) b. If 4 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 79 beats per minute. The probability is (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 sample means, not individuals, the distribution is a normal distribution for any sample size. B. Since the mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size. C. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size. D. Since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size.

Answer 1

a. To find the probability that a randomly selected adult female has a pulse rate less than 79 beats per minute, we need to use the normal distribution. We know that the mean pulse rate (μ) is 75.0 beats per minute and the standard deviation (σ) is 12.5 beats per minute.

We can use the Z-score formula to calculate the Z-score for 79 beats per minute. The Z-score formula is Z = (X - μ) / σ, where X is the value we're interested in.

So, for 79 beats per minute, the Z-score is Z = (79 - 75.0) / 12.5 = 0.32.

Using a Z-table or a calculator, we can find that the probability of a Z-score less than 0.32 is approximately 0.6255. Therefore, the probability that a randomly selected adult female has a pulse rate less than 79 beats per minute is approximately 0.6255.

b. To find the probability that the mean pulse rate of 4 randomly selected adult females is less than 79 beats per minute, we can use the Central Limit Theorem. According to this theorem, when the sample size is large enough (even if it doesn't exceed 30), the distribution of sample means will be approximately normal, regardless of the shape of the original population distribution.

c. The correct answer is C. Since the original population of female pulse rates is normally distributed, the distribution of sample means will also be normally distributed for any sample size. The Central Limit Theorem allows us to use the normal distribution in part (b) because we are dealing with the mean pulse rate of a sample, not the individual pulse rates.