Question

The amount of protein that an individual must consume is different for every person. There are solid theoretical ideas that suggest that the protein requirement will be normally distributed in the population of the United States.The protein requirement is given in terms of the number of grams of good quality protein that must be consumed each day per kilogram body of weight (g P • kg−1 • d−1.) The population mean protein requirement for adults is 0.65 g P • kg−1 • d−1 and the population standard deviation is 0.07 g P • kg−1 • d−1. What proportion of the population have a protein requirement that is less than 0.60 g P • kg-1 • d-1? (Give your answer as a decimal, accurate to three decimal places.)

Answer 1

Answer:

The proportion of the population that have a protein requirement less than 0.60 g P • kg-1 • d-1 is 0.239, that is, 239 persons for every 1000, or simply 23.9% of them.

$\\ 0.239 =\frac{239}{1000}\;or\;23.9\%$

Step-by-step explanation:

From the question, we have the following information:

• The distribution for protein requirement is normally distributed.
• The population mean for protein requirement for adults is $\\ \mu= 0.65 gP*kg^{-1}*d^{-1}$
• The population standard deviation is $\\ \sigma =0.07 gP*kg^{-1}*d^{-1}$

We have here that protein requirements in adults is normally distributed with defined parameters. The question is about the proportion of the population that has a requirement less than $\\ x = 0.60 gP*kg^{-1}*d^{-1}$.

For answering this, we need to calculate a z-score to obtain the probability of the value x in this distribution using a standard normal table available on the Internet or on any statistics book.

z-score

A z-score is expressed as

$\\ z = \frac{x - \mu}{\sigma}$

For the given parameters, we have:

$\\ z = \frac{0.60 - 0.65}{0.07}$

$\\ z = \frac{0.60 - 0.65}{0.07}$

$\\ z = -0.7142857$

Determining the probability

With this value for z at hand, we need to consult a standard normal table to determine what the probability of this value is.

The value for z = -0.7142857 is telling us that the requirement for protein is below the population mean (negative sign indicates this). However, most standard normal tables give a probability that a statistic is less than z and for values greater than the mean (in other words, positive values). To overcome this, we need to take the complement of the probability given for z-score z = 0.7142857, that is, subtract from 1 this probability, which is possible because the normal distribution is symmetrical.

Tables have values for z with two decimal places, then, for z = 0.7142857, we need to rewrite it as z = 0.71. For this value, the standard normal table gives a value of P(z<0.71) = 0.76115.

Therefore, the cumulative probability for values less than x = 0.60 which corresponds to a z-score = -0.7142857 is approximately:

$\\ P(x<0.60) = 1 -P(z<0.71) = 1 - 0.76115 = 0.23885$

$\\ P(x<0.60) = 0.239$ (rounding to three decimal places)

That is, the proportion of the population that have a protein requirement less than 0.60 g P • kg-1 • d-1 is

$\\ 0.239 =\frac{239}{1000}\;or\;23.9\%$

See the graph below. The shaded area is the region that represents the proportion asked in the question.

Answer 2

The amount of protein that an individual must consume is different for every person,there are solid theoretical ideas that suggest that the protein requirement will be normally distributed in the population of the United States.

23.9% of the population that have a protein requirement less than (0.60 gP • kg-1 • d1) or which is 0.239 or 239 persons for every 1000.  

Given:

• (g-P • d−1 • kg−1)the number of grams of good quality protein that must be consumed each day per kilogram body of weight.

• The population mean of protein that are requirement for adults is (0.65 g P • kg−1 • d−1)

• The Population standard deviation is (0.07 g P • kg−1 • d−1).

According to the question, we have the following data here :

• The Distribution for protein requirement = Normally distributed.

• The population mean for protein requirement for adults = The population standard deviation is

Given that the Protein requirements in adults is normally distributed with defined Parameters that requirement is Less .

To solve it, we first need to calculate it by a Z-score concept to obtain the probability of Value of X  In this distribution, we will use a standard Normal table .

What is Z-scope concept ?

Z-score concept is a Numerical measurement which describes the value's relationship to the mean group of values . Z-score is always measure in terms of standard deviation from the mean.

A Z-score is expressed as follows:-

$\rm z= \dfrac{x-\mu }{\sigma}\\\\\rm For \;the\;parameters\;given,\;we\;will\;have\\\\\rm z=\dfrac{0.06-0.65}{0.07}\\\\z= -0.7142857$

How we can determine the probability?

we have now the value of z at hand, we need to check the standard normal table to determine the probability of this value that will be

$\rm z = -0.7142857$

The requirement for protein is the negative population mean .

However,

The most standard normal table give the probability that a statistic is less than z for values larger than the mean positively,to overcome this, we will take the complement of that probability given for z-score

z = 0.7142857, which then subtract from 1,which is possible because the normal distribution is always symmetrical.

Tables consists values for z with two decimal places so,z = 0.7142857, we need to rewrite it as z = 0.71. For this value, the standard normal table gives a value of P(z<0.71) = 0.76115.

Hence,The cumulative probability for values which is less than the value of x = 0.60 which then corresponds to a Z-score = -0.7142857  (approximately).

Therefore, 23.9% of the population that have a protein requirement less than (0.60 gP • kg-1 • d1) or which is 0.239 or 239 persons for every 1000.  

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