Assume that females have pulse rates that are normally distributed with a mean of 74 beats per min and a standard deviation of 12.5 beats per min. If 9 adults are randomly selected, find the probability that they have pulse rates with a mean less than 80 beats per min. Hint: Use the Central limit theorem. O 0.0749 O 1.44 O 0.5 O 0.9251
Assume that females have pulse rates that are normally distributed with a mean of 74 beats per min and a standard deviation of 12.5 beats per min. If 9 adults are randomly selected, find the probability that they have pulse rates with a mean less than 80 beats per min. Hint: Use the Central limit theorem. O 0.0749 O 1.44 O 0.5 O 0.9251
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![**Understanding Probability Using the Central Limit Theorem**
**Problem Statement:**
Assume that females have pulse rates that are normally distributed with a mean of 74 beats per minute (bpm) and a standard deviation of 12.5 bpm.
If 9 adults are randomly selected, find the probability that they have pulse rates with a mean less than 80 bpm.
**Hint:** Use the Central Limit Theorem.
**Options:**
- O 0.0749
- O 1.44
- O 0.5
- O 0.9251
The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean will be normally distributed if the sample size is sufficiently large (usually n > 30), regardless of the shape of the population distribution. However, for smaller sample sizes like 9, the population distribution should be normal.
To solve the problem:
1. **Determine the standard error (SE):**
\[ SE = \frac{\sigma}{\sqrt{n}} \]
where \(\sigma = 12.5\) bpm (standard deviation), and \(n = 9\) (sample size).
\[ SE = \frac{12.5}{\sqrt{9}} = \frac{12.5}{3} = 4.17 \text{ bpm (approximately)} \]
2. **Calculate the z-score:**
\[ Z = \frac{(\bar{x} - \mu)}{SE} \]
where \(\bar{x} = 80\) bpm (sample mean we are comparing to), and \(\mu = 74\) bpm (population mean).
\[ Z = \frac{(80 - 74)}{4.17} = \frac{6}{4.17} \approx 1.44 \]
3. **Find the probability:**
Using standard normal distribution tables or a calculator, find the probability corresponding to \(Z \leq 1.44\).
This yields the approximate probability of 0.9251.
Hence, the correct answer is:
- O 0.9251](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9ed366a-05c2-4728-8c6d-70ad10b89152%2F0d18e533-f360-4347-ad11-9d45ad0f861d%2Fchdbt9h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Probability Using the Central Limit Theorem**
**Problem Statement:**
Assume that females have pulse rates that are normally distributed with a mean of 74 beats per minute (bpm) and a standard deviation of 12.5 bpm.
If 9 adults are randomly selected, find the probability that they have pulse rates with a mean less than 80 bpm.
**Hint:** Use the Central Limit Theorem.
**Options:**
- O 0.0749
- O 1.44
- O 0.5
- O 0.9251
The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean will be normally distributed if the sample size is sufficiently large (usually n > 30), regardless of the shape of the population distribution. However, for smaller sample sizes like 9, the population distribution should be normal.
To solve the problem:
1. **Determine the standard error (SE):**
\[ SE = \frac{\sigma}{\sqrt{n}} \]
where \(\sigma = 12.5\) bpm (standard deviation), and \(n = 9\) (sample size).
\[ SE = \frac{12.5}{\sqrt{9}} = \frac{12.5}{3} = 4.17 \text{ bpm (approximately)} \]
2. **Calculate the z-score:**
\[ Z = \frac{(\bar{x} - \mu)}{SE} \]
where \(\bar{x} = 80\) bpm (sample mean we are comparing to), and \(\mu = 74\) bpm (population mean).
\[ Z = \frac{(80 - 74)}{4.17} = \frac{6}{4.17} \approx 1.44 \]
3. **Find the probability:**
Using standard normal distribution tables or a calculator, find the probability corresponding to \(Z \leq 1.44\).
This yields the approximate probability of 0.9251.
Hence, the correct answer is:
- O 0.9251
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman