Question

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 06255 (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 80 beats per minute. The probability is 08997 (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 307 D OA. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size. OB. Since the distribution is of sample means, not individuals, the distribution is a normal distribution for any sample size OC. Since the distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size OD. Since the mean pulse rate exceeds 30, the distribution of sample means is a normal distribution for any sample size per man Compice parts (a) th Assume that females have pulse rates that are normally distributed with a mean of 75.0 beats per minute and a standard deviation of a 125 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 minuto The probability is tRound to four decimal places as needed.)

Answer 1

If 16 adult females are randomly selected, the probability that they have pulse rates with a mean less than 80 beats per minute can be found using the Central Limit Theorem. According to the Central Limit Theorem, for a large sample size, the distribution of sample means will be approximately normal, regardless of the shape of the original population distribution.

Step-by-step explanation:

b. If 16 adult females are randomly selected, the probability that they have pulse rates with a mean less than 80 beats per minute can be found using the Central Limit Theorem. According to the Central Limit Theorem, for a large sample size, the distribution of sample means will be approximately normal, regardless of the shape of the original population distribution. The mean of the sample means will still be the same as the population mean, which is 75 beats per minute. The standard deviation of the sample means, also known as the standard error, can be calculated as the standard deviation of the population divided by the square root of the sample size. In this case, the standard error would be 125 / sqrt(16) = 31.25. We can then calculate the z-score, which is the number of standard deviations away from the mean. The formula for calculating the z-score is (x - mean) / standard error, where x is the desired value. In this case, we want to find the probability that the sample mean is less than 80, so x = 80. Plugging these values into the formula, we get (80 - 75) / 31.25 = 0.16. We can then look up the corresponding probability in the standard normal distribution table or use a calculator to find that the probability is approximately 0.08997, rounded to four decimal places as requested.

c. The normal distribution can be used in part (b) even though the sample size does not exceed 307 because the Central Limit Theorem states that the distribution of sample means will be approximately normal for any sample size, as long as the original population follows a normal distribution. The distribution of sample means is not affected by the sample size, as long as it is large enough. In this case, the sample size of 16 can be considered large enough to apply the Central Limit Theorem and use the normal distribution.

Answer 2

To find the probability that 40 women have a mean systolic blood pressure greater than 120, one must calculate the z-score using the population mean and standard deviation, along with the sample size, and then use the standard normal distribution to find the corresponding probability.

Step-by-step explanation:

To answer the question about the probability of 40 women having a mean systolic blood pressure greater than 120, we need to use the concept of the sampling distribution of the sample mean. Since we are assuming that the original population of systolic blood pressures is normally distributed, we can continue to use the properties of the normal distribution for the sampling distribution when dealing with sample means, provided the sample size is sufficiently large (usually n > 30 is considered adequate).

The sampling distribution of the sample mean will have a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size, known as the standard error.

We can find the z-score for the sample mean of 120 using the formula z = (Xbar - μ) / (σ/√n), where Xbar is the sample mean we're interested in (120 in this case), μ is the population mean, σ is the population standard deviation, and n is the sample size (40 in this case). The z-score tells us how many standard errors the sample mean is from the population mean. We can then use a standard normal distribution table or a calculator with normal distribution functionalities to find the probability that corresponds to that z-score.