5. Suppose a manufacturer of computer chips is experiencing an average of 2% defective chips.
(a) If a lot contains 1000 chips, what is the probability that more than 25 chips are defective?
HINT. Use the normal approximation to the binomial.
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To find the probability that more than 25 chips are defective, we can use the normal approximation to the binomial distribution. Here's how:
Step 1: Calculate the mean (μ) and standard deviation (σ) of the binomial distribution.
The mean of a binomial distribution is given by μ = n * p, where n is the number of trials and p is the probability of success in each trial.
In this case, the mean is μ = 1000 * 0.02 = 20 chips.
The standard deviation of a binomial distribution is given by σ = sqrt(n * p * (1 - p)).
In this case, the standard deviation is σ = sqrt(1000 * 0.02 * (1 - 0.02)) ≈ 4.36.
Step 2: Convert the problem into a standard normal distribution.
To use the normal approximation, we need to convert the problem into a standard normal distribution. We can do this by standardizing the random variable X using the formula:
Z = (X - μ) / σ
Where Z is the standard normal random variable, X is the random variable from the original distribution, μ is the mean, and σ is the standard deviation.
Step 3: Calculate the probability using the normal distribution.
Now we can calculate the desired probability using the standard normal distribution table or calculator. Since we're interested in the probability of more than 25 defective chips, we subtract the cumulative probability up to 25 from 1.
P(X > 25) = 1 - P(X ≤ 25)
Step 4: Calculate the Z-score for X = 25.
To calculate the Z-score, we plug X = 25, μ = 20, and σ = 4.36 into the formula:
Z = (X - μ) / σ = (25 - 20) / 4.36 ≈ 1.15
Step 5: Look up the cumulative probability for the Z-score.
Using a standard normal distribution table or calculator, we can find the cumulative probability for Z = 1.15 is approximately 0.8749.
Step 6: Calculate the final probability.
Now we can find the probability of X > 25 by subtracting the cumulative probability up to 25 from 1:
P(X > 25) = 1 - 0.8749 ≈ 0.1251
Therefore, the probability that more than 25 chips are defective in a lot of 1000 chips, using the normal approximation to the binomial, is approximately 0.1251 or 12.51%.