Chapter 4: Q103E (page 263)

Blood diamonds. According to Global Research News (March 4, 2014), one-fourth of all rough diamonds produced in the world are blood diamonds, i.e., diamonds mined to finance war or an insurgency. (See Exercise 3.81, p. 200.) In a random sample of 700 rough diamonds purchased by a diamond buyer, let x be the number that are blood diamonds.
a. Find the mean of x.
b. Find the standard deviation of x.
c. Find the z-score for the value x = 200.
d. Find the approximate probability that the number of the 700 rough diamonds that are blood diamonds is less than or equal to 200.

Expert verified

$a . μ = 175 b . σ = 11.4567 c . z = 2.1822 d . The approximate probability is 0.9854$

Step by step solution

According to Global Research News (March 4,2014), one-fourth of all rough diamonds produced in the world are blooddiamonds.

The blood diamond x be a binomial distribution with n = 700 and p = 0.25

$a . μ = n p = 700 \times \frac{1}{4} = 700 \times 0.25 = 175 μ = 175$

Thus, the mean=175.

$b . σ = \sqrt{n p (1 - p)} = \sqrt{700 \times 0.25 \times (1 - 0.25)} = 11.4564 σ = 11.4564$

Therefore, the variance of x is 11.4564.

$c . μ = 175 σ = 11.4564 x = 200$

The z-score is,

$z = \frac{x - μ}{σ} = \frac{200 - 175}{11.4564} = 2.1822 z = 2.1822$

Therefore, the z-score is 2.1822.

$d . P (x \leq 200) = P (x \leq 200) = P (z \leq 2.1822) = 0.9853713 \approx 0.9854 P (x \leq 200) = 0.9854$

Therefore, the approximate probability is 0.9854.

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