An unnoticed mechanical failure has caused one-fourth of a machine shop’s production of 10000 pistol firing pins to be defective. A random sample of 25 firing pins was drawn from the population.

a.) Explain why this random variable has a binomial distribution? (2 points)
b.) What are the mean and the standard deviation of the random variable? (2 points)
c.) Find P(X > 4). Use the binomcdf function on your calculator. (2 points)
d.) Is it appropriate to use the normal approximation to the binomial for this problem? Support your answer with numbers and a test. (3 points)
e.) Using the normal approximation with the continuity correction, calculate the probability that you’ll observe more than 4 defective firing pins in the random sample of 25. Indicate the X-values and the z-scores involved. (3 points)
f.) How does your answer compare with the one you have in part c.)? (2 points)
g.) How do you account for the difference in parts c.) and e.)? (2 points )
h.) Sketch the probability distribution and the area you found using the normal approximation with the continuity correction. (2 points)

a) This random variable has a binomial distribution because it satisfies the following conditions:

1. The experiment consists of repeated trials, which is drawing 25 firing pins from the population.
2. Each trial has two possible outcomes: defective or non-defective firing pins.
3. The probability of success (defective firing pin) remains the same for each trial.
4. The trials are independent, meaning the outcome of one trial does not affect the outcome of another.

b) The mean of a binomial distribution can be calculated using the formula: mean = n * p, where n is the number of trials and p is the probability of success. In this case, n = 25 (the sample size) and p = 1/4 (the probability of a defective firing pin). Therefore, the mean is: mean = 25 * 1/4 = 6.25.

The standard deviation of a binomial distribution can be calculated using the formula: standard deviation = sqrt(n * p * (1 - p)). Plugging in the values, we get: standard deviation = sqrt(25 * 1/4 * (1 - 1/4)) = sqrt(25 * 1/4 * 3/4) = sqrt(18.75) ≈ 4.33.

c) To find P(X > 4), we can use the binomcdf function on a calculator. The "binomcdf" stands for binomial cumulative distribution function and calculates the probability of getting up to a certain number of successes (inclusive). To find P(X > 4), we subtract the cumulative probability up to 4 from 1 (since we want the probability of getting more than 4).

d) To determine if it is appropriate to use the normal approximation to the binomial, we can check for the conditions to be met:
1. The sample size is large enough: The rule of thumb is that both n * p and n * (1 - p) should be greater than 5. In this case, n * p = 25 * 1/4 = 6.25 and n * (1 - p) = 25 * 3/4 = 18.75. Since both values are greater than 5, the sample size is large enough.
2. The distribution is not too skewed: If the distribution is highly skewed, the normal approximation may not be accurate.

e) To calculate the probability using the normal approximation with continuity correction, we need to convert the binomial distribution to a normal distribution. The formula for the z-score is: z = (x - mean) / standard deviation, where x is the number of successes. In this case, x is greater than 4, so we need to calculate P(X > 4).

f) We will compare the result from part c and part e to see how they differ.

g) The difference between part c and part e can be attributed to different methods of calculation. Part c used the binomcdf function to directly calculate the probability from the binomial distribution, while part e used the normal approximation with continuity correction to approximate the binomial distribution as a normal distribution.

h) Sketch the probability distribution showing the x-values (number of defective firing pins) on the x-axis and the probabilities on the y-axis. Indicate the area calculated using the normal approximation with the continuity correction.