4. John Rengel is the Quality Assurance Supervisor for Vino Supremo Vinyards. He knows that 10 percent of each box of corks is undersized. (6 pts)

a) If he were to randomly select 120 corks from the next box, then how many of these corks would John expect to be undersized?
b) If he were to randomly select 120 corks from each box, then what would John calculate as the standard error of the number of undersized corks?
c)What is the probability that John will find 15 or more corks defective in a randomly selected box?

a) 12

b) 12
c) 0

a) If he were to randomly select 120 corks from the next box, then how many of these corks would John expect to be undersized?



E(x) h*p
= 12


b) If he were to randomly select 120 corks from each box, then what would John calculate as the standard error of the number of undersized corks?


Standard Error = σ
σ ═ SQRT(0.1*(1-0.1)/120)
0.027386128


c) What is the probability that John will find 15 or more corks defective in a randomly selected box?

cumulative binomial distribution
Probabilty P (X ≥ 15)
=1-BINOM.DIST(number_s,trials,probability_s,cumulative)
0.218163357
21.8%

a) To calculate the number of undersized corks John would expect to find when randomly selecting 120 corks from the next box, he can use the concept of probability.

Since 10% of each box of corks is undersized, it means that out of 100 corks in a box, 10 would be undersized.

Therefore, the expected number of undersized corks in the next box would be 10% of 120, which is 0.10 x 120 = 12 undersized corks.

So, John would expect to find 12 undersized corks when randomly selecting 120 corks from the next box.

b) The standard error is a measure of how much the results of a sample may vary from the true population value. To calculate the standard error of the number of undersized corks, John can use the formula:

Standard Error = Square root [ (probability of success) x (probability of failure) / sample size ]

In this case, the probability of success is 0.10 (as 10% of corks are undersized) and the probability of failure is 0.90 (as 90% of corks are not undersized). The sample size is 120.

Standard Error = Square root [ (0.10) x (0.90) / 120 ]
Standard Error = Square root [ 0.009 / 120 ]
Standard Error = Square root [ 0.000075 ]
Standard Error ≈ 0.0087 (rounded to 4 decimal places)

Therefore, the standard error of the number of undersized corks is approximately 0.0087.

c) To calculate the probability of finding 15 or more defective corks in a randomly selected box, John can use the binomial probability formula:

P(x ≥ k) = 1 - P(x < k)

Where:
P(x ≥ k) is the probability of finding k or more defective corks
P(x < k) is the probability of finding less than k defective corks

In this case, k is 15, and we need to find the probability of finding 15 or more defective corks.

To find P(x < k), we sum the probabilities of finding 0, 1, 2, ..., k-1 defective corks. The formula for calculating the probability of finding k defective corks is given as:

P(x = k) = (nCk) * (p^k) * (q^(n-k))

Where:
n is the sample size (120 corks)
k is the number of defective corks (15)
p is the probability of success (0.10)
q is the probability of failure (0.90)

Now, let's calculate P(x ≥ 15) using the formula:

P(x < 15) = P(x = 0) + P(x = 1) + ... + P(x = 14)

P(x < 15) = [(120C0) * (0.10^0) * (0.90^120)] + [(120C1) * (0.10^1) * (0.90^119)] + ... + [(120C14) * (0.10^14) * (0.90^106)]

Calculating this sum may take several steps and some calculations. To find an accurate result, it is recommended to use software, a calculator, or a statistical table to calculate the binomial probabilities for each step and then sum them up.

Once the probabilities are calculated, subtract the probability of finding less than 15 defective corks from 1 to obtain the probability of finding 15 or more defective corks.

P(x ≥ 15) = 1 - P(x < 15)

Using the calculated values, you can find the exact probability. However, it is not possible to provide an exact value without calculating it step-by-step using software or a calculator.