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:

$\\ P(x$

$\\ P(x$ (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.