1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Elena-2011 [213]
3 years ago
5

An example of an early application of statistics was in the year 1817. A study of chest circumference among a group of Scottish

men exhibited an approximately normal distribution. Their chest circumferences ranged from 33 to 48 inches, with a mean chest measurement of 40 inches and a standard deviation of 2 inches. Make a normal bell curve, then use the Empirical Rule to help you understand the distribution of chest circumferences in the study.
a) What range of chest measurements contains the 58% which fall closest to the mean?


b) What would you expect the chest measurements to be fore the 2.5% of the men with the smallest chest measurements?


c) What would you expect to be the smallest and largest chest measurements of the "middle" 95% of the men?


d) What would you expect to be the measurements of the 16% of the men with the largest chests in the population

Mathematics
1 answer:
otez555 [7]3 years ago
8 0

Answer:

Step-by-step explanation:

Hello!

The variable of interest is X: chest circumference of a Scottish man.

X≈N(μ;δ²)

μ= 40 inches

δ= 2 inches

The empirical rule states that

68% of the distribution lies within one standard deviation of the mean: μ±δ= 0.68

95% of the distribution lies within 2 standard deviations of the mean: μ±2δ= 0.95

99% of the distribution lies within 3 standard deviations of the mean: μ±3δ= 0.99

a)

The 58% that falls closest to the mean can also be referred to as the middle 58% of the distribution, assuming that both values are equally distant from the mean.

P(a≤X≤b)= 0.58

If 1-α= 0.58, then the remaining proportion α= 0.42 is divided in two equal tails α/2= 0.21.

The accumulated proportion until "a" is 0.21 and the accumulated proportion until "b" is 0.21 + 0.58= 0.79 (See attachment)

P(X≤a)= 0.21

P(X≤b)= 0.79

Using the standard normal distribution, you can find the corresponding values for the accumulated probabilities, then using the information of the original distribution:

P(Z≤zᵃ)= 0.21

zᵃ= -0.806

P(Z≤zᵇ)= 0.79

zᵇ= 0.806

Using the standard normal distribution Z= (X-μ)/δ you "transform" the values of Z to values of chest circumference (X):

zᵃ= (a-μ)/δ

zᵃ*δ= a-μ

a= (zᵃ*δ)+μ

a= (-0.806*2)+40= 38.388

and

zᵇ= (b-μ)/δ

zᵇ*δ= b-μ

b= (zᵇ*δ)+μ

b= (0.806*2)+40= 41.612

58% of the chest measurements will be within 38.388 and 41.612 inches.

b)

The measurements of the 2.5% men with the smallest chest measurements, can also be interpreted as the "bottom" 2.5% of the distribution, the value that separates the bottom 2.5% of the distribution from the 97.5%, symbolically:

P(X≤b)= 0.025 (See attachment)

Now you have to look under the standard normal distribution the value of z that accumulates 0.025 of the distribution:

P(Z≤zᵇ)= 0.025

zᵇ= -1.960

Now you reverse the standardization to find the value of chest circumference:

zᵇ= (b-μ)/δ

zᵇ*δ= b-μ

b= (zᵇ*δ)+μ

b= (-1.960*2)+40= 36.08

The chest measurement of the 2.5% smallest chest measurements is 36.08 inches.

c)

Using the empirical rule:

95% of the distribution lies within 2 standard deviations of the mean: μ±2δ= 0.95

(μ-2δ) ≤ Xc ≤ (μ+2δ)=0.95 ⇒ (40-4) ≤ Xc ≤ (40+4)= 0.95 ⇒ 36 ≤ Xc ≤ 44= 0.95

d)

The measurements of the 16% of the men with the largest chests in the population or the "top" 16% of the distribution:

P(X≥d)= 0.16

P(X≤d)= 1 - 0.16

P(X≤d)= 0.84

First, you look for the value that accumulates 0.84 of probability under the standard normal distribution:

P(Z≤zd)= 0.84

zd= 0.994

Now you reverse the standardization to find the value of chest circumference:

zd= (d-μ)/δ

zd*δ= d-μ

d= (zd*δ)+μ

d= (0.994*2)+40= 41.988

The measurements of the 16% of the men with larges chess are at least 41.988 inches.

I hope this helps!

You might be interested in
What is the distance between the points (-5, 10) and (-5, 2)?
Alona [7]
8 units hope I helped
7 0
3 years ago
PLSSS HELP IF YOU TURLY KNOW THISS
hjlf

Refer the attachment

Make me as brain liest

4 0
2 years ago
Read 2 more answers
Show the work please
frosja888 [35]

Answer: x = 9/2

Step-by-step explanation: 24 = 4x + 6

7 0
3 years ago
Read 2 more answers
Is this is correct? I’m confused you can help me at the last one to if you want
Wittaler [7]

Answer:

its correct

Step-by-step explanation:

because if u take away 1 and add for take away 2 and get 3

4 0
2 years ago
Help me get the answer
drek231 [11]
This is hard- DHJDHSJS
3 0
3 years ago
Other questions:
  • Mr. Beachem is selling 88.7 ice cream cakes. 34 were bought, and 8 were stolen. How many ice cream cakes are there?
    6·1 answer
  • 2.8x = -9.24 <br>A. 12.04<br>B. -0.3030<br>C. 12.04<br>D. -3.3​
    13·2 answers
  • A bird flies 1 mile north, then 1 mile east. how far is it from where it started?
    15·1 answer
  • The surface areas of two similar figures are 25 in2 and 36 in2. If the volume of the smaller figure is 250 in3, what is the volu
    14·2 answers
  • The product of -5 and the sum of X and eight decreased by the product of three and X
    11·1 answer
  • How do i solve 2+49x4+x
    15·1 answer
  • Raj used 4 3/4 cups of sugar for each batch of muffins. He made 3 1/2 batches. Raj used 16 5/8 cups of sugar altogether.
    13·2 answers
  • Evaluate 11.3p + 10.7r when p = 3 and r = 4
    8·1 answer
  • PLEASE HELP!!!
    14·1 answer
  • Tony is flying from Los Angeles to Boston. His plane ticket costs $605 and the total flight distance is 3,052 miles. What is the
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!