A chemical engineer monitors a dyeing proccess for polyester

A chemical engineer monitors a dyeing proccess for polyester yarn used in clothig by comparing a sample of yarn against a standard color chart. He accepts or rejects the entire deing batch based on the samples results. Historically, this proccess averages 5% reject batches. Each shift, the proccess dyes eight batches. Assume the next shift forms on a random sample.

A) find the probability that at least 7 batches are accepted

b) The probability that at least one batch is rejected

c) Find the expected number of batches accepted, the variance, and the standard deviation

d) find the expected number of batches rejected, the variance, and the standard deviation

Here,

p(reject) = 0.05
p(accept) = 1-0.05 = 0.95

Note that P(at least x) = 1 - P(at most x - 1).

Using a cumulative binomial distribution table or technology, matching

n = number of trials =    8
p = the probability of a success =    0.95
x = our critical value of successes =    7

Then the cumulative probability of P(at most x - 1) from a table/technology is

P(at most   6   ) =    0.05724465

Thus, the probability of at least   7   successes is

P(at least   7   ) =    0.94275535 [ANSWER]

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Note that P(at least x) = 1 - P(at most x - 1).

Using a cumulative binomial distribution table or technology, matching

n = number of trials =    8
p = the probability of a success =    0.05
x = our critical value of successes =    1

Then the cumulative probability of P(at most x - 1) from a table/technology is

P(at most   0   ) =    0.663420431

Thus, the probability of at least   1   successes is

P(at least   1   ) =    0.336579569 [ANSWER]

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E(x) = n p = 8*0.95 = 7.6 [ANSWER]

Var(x) = np(1-p) = 8*0.95*(1-0.95) = 0.38 [ANSWER]

s(x) = sqrt(Var(x)) = 0.6164414 [ANSWER]

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E(x) = n p = 8*0.05 = 0.4 [ANSWER]

Var(x) = np(1-p) = 8*0.05*(1-0.05) = 0.38 [ANSWER]

s(x) = sqrt(Var(x)) = 0.6164414 [ANSWER]