Suppose I work on a factory line doing quality control work. Since I've been working here for such a long time, I am fairly confident that 15% of all items (independent of all other items) need to be sent back to be re-worked. These items take a fairly long time to check properly, but it is the same set of steps every time, so I always check 25 items every shift. Let X be the number of items sent back to be re-worked during a given shift. What is the distribution of X? Give the name of the distribution and the appropriate parameters. What is the mean and variance of this distribution? Give your answer as a number, but include the formulas (or logic) used. What is the probability I send back less than 4 items to be re-worked during a single shift? What is the probability I send back exactly 3 items to be re-worked during a single shift?

Step-by-step explanation:

a) It is a binomial distribution.

n = 25

p = 0.15

b) mean = np = 25 * 0.15 = 3.75

   variance = np(1 - p)

                 = 25 * 0.15 * (1 - 0.15)

                 = 3.1875

c) P(X = x) = nCx * px * (1 - p)n - x

P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

            = 25C0 * (0.15)^0 * (0.85)^25 + 25C1 * (0.15)^1 * (0.85)^24 + 25C2 * (0.15)^2 * (0.85)^23 + 25C3 * (0.15)^3 * (0.85)^22 = 0.4711

d) P(X = 3) = 25C3 * (0.15)^3 * (0.85)^22 = 0.2174

so firstly, we need to isolate the y variables to be able to solve these inequalities. firstly, add 0.5x on both sides of the first inequality and subtract 1.5x on both sides of the second inequality:

now since the slope is positive for the first inequality, this means that the line going upwards belongs to the first inequality, and the line going downwards belongs to the second inequality.

next, since y ≥ in the first inequality, this means that the shaded region will be above the first inequality's line, thus shading regions a and b.

next, since y ≤ in the second inequality, this means that the shaded region is going to be below this line, thus shading regions a and c.

since both lines shade region a, this means that region a is the solution.

step-by-step explanation:

mark brainliest!