Assume that females have pulse rates that are normally distributed with a mean of μ=73.0 beats per minute and a standard deviation of σ= 12.5 beats per minute b) If 16 adult females are randomly selected, find the probability that they have pulse rates with a mean between 66 beats per minute and 80 beats per minute The probability is______(Round to 4 decimal places as needed) **Understanding the Normal Distribution in Sampling****Question:**Why can the normal distribution be used in part (b), even though the sample size does not exceed 30?**Options:**A. Since the mean pulse rate exceeds 30, the distribution of sample means 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 distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size.In this question, we are exploring the conditions under which the normal distribution can be applied to sample means, especially when the sample size is relatively small (30 or fewer). Each option presents a different rationale that needs to be evaluated for accuracy concerning statistical theory and practices.- **Option A** mentions an incorrect reasoning that ties the mean pulse rate with the distribution type.- **Option B** correctly identifies that if the original population is normally distributed, the sample means will also follow a normal distribution regardless of the sample size.- **Option C** correctly points out that the focus on sample means (rather than individual values) justifies the assumption of normality under certain conditions.- **Option D** incorrectly implies that the distribution of individual values being normal directly applies to the use of sample means in small samples.This question illustrates essential concepts in statistics, particularly around the Central Limit Theorem and sample distributions.**Graphical Explanation (Not Present):**If provided, a graph showcasing the normal distribution with a sample mean would typically illustrate how the distribution curve remains bell-shaped regardless of the sample size, provided the population is normally distributed. This reinforces the understanding of how sample means behave under normal distribution assumptions.

Answered: Assume that females have pulse rates…

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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Assume that females have pulse rates that are normally distributed with a mean of μ=73.0 beats per minute and a standard deviation of σ= 12.5 beats per minute

b) If 16 adult females are randomly selected, find the probability that they have pulse rates with a mean between 66 beats per minute and 80 beats per minute

The probability is______(Round to 4 decimal places as needed)

**Understanding the Normal Distribution in Sampling**

**Question:**
Why can the normal distribution be used in part (b), even though the sample size does not exceed 30?

**Options:**
A. Since the mean pulse rate exceeds 30, the distribution of sample means 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 distribution is of individuals, not sample means, the distribution is a normal distribution for any sample size.

In this question, we are exploring the conditions under which the normal distribution can be applied to sample means, especially when the sample size is relatively small (30 or fewer). Each option presents a different rationale that needs to be evaluated for accuracy concerning statistical theory and practices.

- **Option A** mentions an incorrect reasoning that ties the mean pulse rate with the distribution type.
- **Option B** correctly identifies that if the original population is normally distributed, the sample means will also follow a normal distribution regardless of the sample size.
- **Option C** correctly points out that the focus on sample means (rather than individual values) justifies the assumption of normality under certain conditions.
- **Option D** incorrectly implies that the distribution of individual values being normal directly applies to the use of sample means in small samples.

This question illustrates essential concepts in statistics, particularly around the Central Limit Theorem and sample distributions.

**Graphical Explanation (Not Present):**
If provided, a graph showcasing the normal distribution with a sample mean would typically illustrate how the distribution curve remains bell-shaped regardless of the sample size, provided the population is normally distributed. This reinforces the understanding of how sample means behave under normal distribution assumptions.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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