Basic Practice of Statistics (Instructor's)
Basic Practice of Statistics (Instructor's)
8th Edition
ISBN: 9781319057923
Author: Moore
Publisher: MAC HIGHER
bartleby

Concept explainers

bartleby

Videos

Question
Book Icon
Chapter 15, Problem 15.32E

(a)

To determine

To find: The approximate distribution of x¯ according to the central limit theorem.

(a)

Expert Solution
Check Mark

Answer to Problem 15.32E

The approximate distribution of x¯ is normal with mean μ=8.8beatsperfiveseconds and the standard deviation is σn=0.2041beatsperfiveseconds .

Explanation of Solution

Given info:

The data shows that the mean heart rate (μ) is 8.8 beats per five seconds and the standard deviation (σ) is 1 beat per five seconds. The distribution of heart rates is not normal.

Calculation:

Central limit theorem:

It states that as the sample size n becomes large, the sampling distribution of x¯ follows normal distribution with mean μ and standard deviation σn .

Mean of the Sampling distribution of x¯ :

The mean of the sampling distribution of x¯ is μ . That is, 8.8.

Thus, the mean of the average heart rate x¯ is 8.8 beats per five seconds.

Standard deviation of the Sampling distribution of x¯ :

The standard deviation can be calculated by using the formula σn .

Substitute 1 for σ and 24 for n

σn=124=14.8990=0.2041

Thus, the standard deviation of x¯ is 0.2041 beats per five seconds.

Hence, by the central limit theorem,the approximate distribution of x¯ is normal with mean 8.8beatsperfiveseconds and standard deviation is 0.2041beatsperfiveseconds .

(b)

To determine

To find: The approximate probability that x¯ less than 8.

(b)

Expert Solution
Check Mark

Answer to Problem 15.32E

The approximate probability of x¯ less than 8 is 0.

Explanation of Solution

Calculation:

STATE:

The mean heart rate (μ) is 8.8 beats per five seconds and the standard deviation (σ) is 1 beat per five seconds. What is the approximate probability that x¯ is less than 8?

PLAN:

The distribution of heart rates is not normal. Then, the central limit theorem would be used to find the distribution of mean heart rate.

SOLVE:

Let X be the number of heart beats per five seconds and x¯ be the mean number of beats per five seconds.

From the part a, x¯ is normal with mean μ=8.8beatsperfiveseconds and the standard deviation is σn=0.2041beatsperfiveseconds .

Then,

P(x¯<8)=P(x¯μσn<88.8124)=P(Z<0.80.2041)=P(Z<3.92)0[FromTableA:StandardNormalCumulativeProportions]

Thus, the approximate probability of x¯ less than 8 is 0.

CONCLUSION:

The approximate probability of x¯ less than 8 is 0.

(c)

To determine

To find: The approximate probability that the heart rate of a runner is less than 100 beats per minute.

(c)

Expert Solution
Check Mark

Answer to Problem 15.32E

The approximate probability that the heart rate of a runner is less than 100 beats per minute is 0.0107.

Explanation of Solution

Calculation:

The number of five seconds in one minute is 605=12 .

The mean heart rate for five seconds is 10012=8.33beatsperfiveseconds .

Then,

P(x¯<8)=P(x¯μσn<8.338.8124)=P(Z<0.470.2041)=P(Z<2.30)=0.0107[FromTableA:StandardNormalCumulativeProportions]

Thus, the approximate probability that the heart rate of a runner is less than 100 beats per minute is 0.0107.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
Calculate the 90% confidence interval for the population mean difference using the data in the attached image. I need to see where I went wrong.
Microsoft Excel snapshot for random sampling: Also note the formula used for the last column 02 x✓ fx =INDEX(5852:58551, RANK(C2, $C$2:$C$51)) A B 1 No. States 2 1 ALABAMA Rand No. 0.925957526 3 2 ALASKA 0.372999976 4 3 ARIZONA 0.941323044 5 4 ARKANSAS 0.071266381 Random Sample CALIFORNIA NORTH CAROLINA ARKANSAS WASHINGTON G7 Microsoft Excel snapshot for systematic sampling: xfx INDEX(SD52:50551, F7) A B E F G 1 No. States Rand No. Random Sample population 50 2 1 ALABAMA 0.5296685 NEW HAMPSHIRE sample 10 3 2 ALASKA 0.4493186 OKLAHOMA k 5 4 3 ARIZONA 0.707914 KANSAS 5 4 ARKANSAS 0.4831379 NORTH DAKOTA 6 5 CALIFORNIA 0.7277162 INDIANA Random Sample Sample Name 7 6 COLORADO 0.5865002 MISSISSIPPI 8 7:ONNECTICU 0.7640596 ILLINOIS 9 8 DELAWARE 0.5783029 MISSOURI 525 10 15 INDIANA MARYLAND COLORADO
Suppose the Internal Revenue Service reported that the mean tax refund for the year 2022 was $3401. Assume the standard deviation is $82.5 and that the amounts refunded follow a normal probability distribution. Solve the following three parts? (For the answer to question 14, 15, and 16, start with making a bell curve. Identify on the bell curve where is mean, X, and area(s) to be determined. 1.What percent of the refunds are more than $3,500? 2. What percent of the refunds are more than $3500 but less than $3579? 3. What percent of the refunds are more than $3325 but less than $3579?
Knowledge Booster
Background pattern image
Statistics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
MATLAB: An Introduction with Applications
Statistics
ISBN:9781119256830
Author:Amos Gilat
Publisher:John Wiley & Sons Inc
Text book image
Probability and Statistics for Engineering and th...
Statistics
ISBN:9781305251809
Author:Jay L. Devore
Publisher:Cengage Learning
Text book image
Statistics for The Behavioral Sciences (MindTap C...
Statistics
ISBN:9781305504912
Author:Frederick J Gravetter, Larry B. Wallnau
Publisher:Cengage Learning
Text book image
Elementary Statistics: Picturing the World (7th E...
Statistics
ISBN:9780134683416
Author:Ron Larson, Betsy Farber
Publisher:PEARSON
Text book image
The Basic Practice of Statistics
Statistics
ISBN:9781319042578
Author:David S. Moore, William I. Notz, Michael A. Fligner
Publisher:W. H. Freeman
Text book image
Introduction to the Practice of Statistics
Statistics
ISBN:9781319013387
Author:David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:W. H. Freeman
The Shape of Data: Distributions: Crash Course Statistics #7; Author: CrashCourse;https://www.youtube.com/watch?v=bPFNxD3Yg6U;License: Standard YouTube License, CC-BY
Shape, Center, and Spread - Module 20.2 (Part 1); Author: Mrmathblog;https://www.youtube.com/watch?v=COaid7O_Gag;License: Standard YouTube License, CC-BY
Shape, Center and Spread; Author: Emily Murdock;https://www.youtube.com/watch?v=_YyW0DSCzpM;License: Standard Youtube License