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 0.70 g P • kg⁻¹ • d⁻¹? (Give your answer as a decimal, accurate to three decimal places.)

Find the probability that a randomly selected person will have a protein requirement that is between 0.60 and 0.70 g P • kg⁻¹ • d⁻¹. (Hint: Remember that to find the area between two values, shade to the left of each Z-Score you calculate and then subtract the smaller area from the larger area. Make sure your answer is positive. This is the probability that a randomly selected observation will lie between the two values. Give your answer as a decimal, accurate to three decimal places.)

Find the 35th percentile for the protein requirement for adults. Round your answer to three decimal places.

Find the 80th percentile for the protein requirement for adults. Round your answer to three decimal places.

Answer 1

Consider the random variable

X : Protein requirements for adults.

mu : Mean protein requirement for adults. = 0.65

sigma : population standard deviation = 0.07.

X ~ N ( mu = 0.65, sigma² = 0.07²)

i) Required probability = P ( X < 0.70)

From normal probability table

P ( Z < 0.7143) = 0.7625

Proportion of the population have a protein requirement is less than 0.70 = 76.25%

ii) Required Probability = P ( 0.60 < X < 0.70)

= P ( -0.7143 < Z < 0.7143)

= P ( Z < 0.7143) - P ( Z < - 0.7143)

From normal probability table

P ( Z < 0.7143) = 0.7625 and P ( Z < -0.7143) = 0.2375.

Required Probability = 0.7625 - 0.2375 = 0.5275

P ( Selected person will have a protein requirement between 0.60 and 0.70 ) = 0.5275.

iii) Let a be 35^th percentile for protein requirement for adults.

P ( X < a ) = 0.35

---------------(I)

From normal probability table

P ( Z < -0.3853) = 0.35 -----------------(II)

From (I) and (II)

35^th percentile for protein requirement for adults = 0.6230

iv) Let b be 80^th percentile for protein requirement for adults.

P ( X < b ) = 0.35

---------------(III)

From normal probability table

P ( Z < 0.8416) = 0.80-----------------(IV)

From (III) and (IV)

80^th percentile for protein requirement for adults = 0.7089