This problem involves finding probabilities related to a normal distribution. The given information is:

• Mean ($\mu$) = 10 products per hour
• Standard deviation ($\sigma$) = 1.5 products per hour

We assume the production follows a normal distribution. To solve for the probabilities, we will use the standard normal distribution $Z$-scores.

The formula for the $Z$-score is:

$Z = \frac{X - \mu}{\sigma}$

where:

• $X$ is the value of interest,
• $\mu$ is the mean,
• $\sigma$ is the standard deviation.

(a) Probability of producing less than 12.8 products per hour

We need to find $P(X < 12.8)$.

1. First, compute the $Z$-score for $X = 12.8$: $Z = \frac{12.8 - 10}{1.5} = \frac{2.8}{1.5} \approx 1.87$

2. Now, we find the probability corresponding to $Z = 1.87$. Using a standard normal table, the cumulative probability for $Z = 1.87$ is approximately 0.9693.

Therefore, the probability of producing less than 12.8 products per hour is:

$P(X < 12.8) \approx 0.9693$

(b) Probability of producing more than 12 products per hour

We need to find $P(X > 12)$.

1. First, compute the $Z$-score for $X = 12$: $Z = \frac{12 - 10}{1.5} = \frac{2}{1.5} \approx 1.33$

2. Now, we find the cumulative probability for $Z = 1.33$ from the standard normal table, which is approximately 0.9082. Since we want the probability for $X > 12$, we subtract this value from 1:

$P(X > 12) = 1 - 0.9082 = 0.0918$

Therefore, the probability of producing more than 12 products per hour is:

$P(X > 12) \approx 0.0918$

Summary:

• (a) $P(X < 12.8) \approx 0.9693$
• (b) $P(X > 12) \approx 0.0918$

Would you like to see the details of these calculations, or any further clarification?

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Here are 5 related questions to expand your understanding:

1. How do we interpret the $Z$-score in a normal distribution?
2. What is the probability of producing between 9 and 12 products per hour?
3. How would the probabilities change if the standard deviation increased to 2?
4. Can we calculate the probability of producing exactly 10 products per hour?
5. How do we apply the concept of the central limit theorem to production rates?

Tip: Always visualize normal distribution problems using a bell curve to better understand areas under the curve and probabilities.