An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are independently and normally distributed with a mean of 120 cm and a standard deviation of 4.7 cm.

Find the probability that if 3 are randomly selected, all 3 have lengths that exceed 122 cm.

To find the probability that all 3 subcomponents have lengths exceeding 122 cm, we need to calculate the probability of a single subcomponent having a length exceeding 122 cm, and then raise that probability to the power of 3 since we are selecting 3 subcomponents.

Step 1: Calculate the Z-score for 122 cm
We can use the formula: Z = (X - μ) / σ
Where X is the value we want to find the Z-score for, μ is the mean, and σ is the standard deviation.

Z = (122 - 120) / 4.7
Z = 2 / 4.7
Z ≈ 0.426

Step 2: Find the cumulative probability for Z = 0.426
We need to find the area under the standard normal curve to the right of Z = 0.426, which represents the probability of a subcomponent having a length exceeding 122 cm. We can use a standard normal distribution table or a calculator to find this cumulative probability.

Using a standard normal distribution table or a calculator, we find that the cumulative probability for Z = 0.426 is approximately 0.6652.

Step 3: Calculate the probability for all 3 subcomponents having lengths exceeding 122 cm
Since the lengths of the subcomponents are independently distributed, we can simply multiply the probability of a single subcomponent having a length exceeding 122 cm by itself 3 times.

P(all 3 subcomponents > 122 cm) = 0.6652^3
P(all 3 subcomponents > 122 cm) ≈ 0.296

Therefore, the probability that if 3 subcomponents are randomly selected, all 3 have lengths that exceed 122 cm is approximately 0.296, or 29.6%.

To find the probability that all 3 selected subcomponents have lengths that exceed 122 cm, we can use the concept of independent events.

1. First, we need to calculate the probability that a single subcomponent has a length that exceeds 122 cm.
Let's denote this probability as P(X > 122), where X represents the length of a subcomponent. We can find this probability using the standard normal distribution.

Using the z-score formula, we can calculate the z-score for a subcomponent length of 122 cm:

z = (x - μ) / σ
= (122 - 120) / 4.7
= 2 / 4.7
≈ 0.4255

Next, we can find the probability using the z-table or a statistical calculator. P(Z > 0.4255) is approximately 0.3357.

2. Since each subcomponent is independently selected, the probability that all 3 selected subcomponents have lengths that exceed 122 cm can be calculated by multiplying the individual probabilities together.

P(all 3 exceed 122 cm) = P(X > 122) * P(X > 122) * P(X > 122)
= 0.3357 * 0.3357 * 0.3357
≈ 0.0380

Therefore, the probability that all 3 randomly selected subcomponents have lengths that exceed 122 cm is approximately 0.0380 or 3.80%.

Use z-score formula:

z = (x - mean)/sd

z = (122 - 120)/4.7 = ?

Once you have the z-score, check a z-table for the probability (keep in mind that you are looking for the probability exceeding 122 cm).

After you have the probability from the table, use a normal approximation to the binomial distribution.

Formulas:
mean = np = 3 * p
sd = √npq = √(3 * p * q)
Note: p = probability from z-table; q = 1 - p

Use z-scores again; this time use 3 for x, the mean calculated above, and the standard deviation calculated above.

Once you have this z-score, determine the probability using a z-table once again.

I hope this will help get you started.