Answer:
(a) The distribution of <em>X</em> is <em>N</em> (9.8, 0.8²).
(b) The probability that an American consumes between 8.8 and 9.9 grams of sodium per day is 0.4461.
(c) The middle 30% of American men consume between 9.5 grams to 10.1 grams of sodium.
Step-by-step explanation:
The random variable <em>X</em> is defined as the amount of sodium consumed.
The random variable <em>X</em> has an average value of, <em>μ</em> = 9.8 grams.
The standard deviation of <em>X</em> is, <em>σ</em> = 0.8 grams.
(a)
It is provided that the sodium consumption of American men is normally distributed.
The random variable <em>X</em> follows a normal distribution with parameters <em>μ</em> = 9.8 grams and <em>σ</em> = 0.8 grams.
Thus, the distribution of <em>X</em> is <em>N</em> (9.8, 0.8²).
(b)
If X ~ N (µ, σ²), then
, is a standard normal variate with mean, E (Z) = 0 and Var (Z) = 1. That is, Z ~ N (0, 1).
To compute the probability of Normal distribution it is better to first convert the raw score (<em>X</em>) to <em>z</em>-scores.
Compute the probability that an American consumes between 8.8 and 9.9 grams of sodium per day as follows:
![P(8.8](https://tex.z-dn.net/?f=P%288.8%3CX%3C9.9%29%20%3DP%28%5Cfrac%7B8.8-9.8%7D%7B0.8%7D%3C%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B9.9-9.8%7D%7B0.8%7D%29)
![=P(-1.25](https://tex.z-dn.net/?f=%3DP%28-1.25%3CZ%3C0.125%29%5C%5C%3DP%28Z%3C0.125%29-P%28Z%3C-1.25%29%5C%5C%3D0.5517-0.1056%5C%5C%3D0.4461)
Thus, the probability that an American consumes between 8.8 and 9.9 grams of sodium per day is 0.4461.
(c)
The probability representing the middle 30% of American men consuming sodium between two weights is:
![P(x_{1}](https://tex.z-dn.net/?f=P%28x_%7B1%7D%3CX%3Cx_%7B2%7D%29%3D0.30%5C%5C%5CRightarrow%20P%28-z%3CZ%3Cz%29%3D0.30)
Compute the value of <em>z</em> as follows:
![P(-z](https://tex.z-dn.net/?f=P%28-z%3CZ%3Cz%29%3D0.30%5C%5CP%28Z%3Cz%29-P%28Z%3C-z%29%3D0.30%5C%5CP%28Z%3Cz%29-%5B1-P%28Z%3Cz%29%5D%3D0.30%5C%5C2P%28Z%3Cz%29-1%3D0.30%5C%5CP%28Z%3Cz%29%3D0.65)
The value of <em>z</em> for P (Z < z) = 0.65 is 0.39.
Compute the value of <em>x</em>₁ and <em>x</em>₂ as follows:
![z=\frac{x_{2}-\mu}{\sigma}\\0.39=\frac{x_{1}-9.8}{0.8}\\x_{1}=9.8+(0.39\times 0.8)\\x_{1}=10.112\\x_{1}\approx10.1](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bx_%7B2%7D-%5Cmu%7D%7B%5Csigma%7D%5C%5C0.39%3D%5Cfrac%7Bx_%7B1%7D-9.8%7D%7B0.8%7D%5C%5Cx_%7B1%7D%3D9.8%2B%280.39%5Ctimes%200.8%29%5C%5Cx_%7B1%7D%3D10.112%5C%5Cx_%7B1%7D%5Capprox10.1)
Thus, the middle 30% of American men consume between 9.5 grams to 10.1 grams of sodium.