91Ó°ÊÓ

The normal distribution, often called the bell curve due to its shape, is vitally important in statistics. Imagine it as a smooth hill centered around a particular value—this is the mean. In this scenario, both cork-producing machines have different mean diameters and standard deviations, which dictate the spread or "width" of their hills.

• The first machine has a mean of 3 cm and a standard deviation of 0.1 cm, meaning its diameters are spread more widely around the mean.
• The second machine has a mean of 3.04 cm with a smaller standard deviation of 0.02 cm, making its distribution taller and narrower.

For a cork to be considered "acceptable," its diameter must lie between 2.9 cm and 3.1 cm. This means we’re looking at how much area—or probability—falls within this range under each machine’s bell curve.

The Z-score is a wonderfully useful tool that tells us how far away a particular value is from the mean, measured in standard deviations. To judge the acceptability of corks, Z-scores help in comparing differences in measurements, by converting real-world values into a standard normal distribution language.

• For the first machine:
  - Calculating for 2.9 cm: \[z_1 = \frac{2.9 - 3.0}{0.1} = -1\]
  - Calculating for 3.1 cm: \[z_2 = \frac{3.1 - 3.0}{0.1} = 1\]
• For the second machine:- Calculating for 2.9 cm: \[z_1 = \frac{2.9 - 3.04}{0.02} = -7\]
  - Calculating for 3.1 cm: \[z_2 = \frac{3.1 - 3.04}{0.02} = 3\]

These calculations allow us to use Z-tables to determine the probability that a cork’s diameter lies between these values, representing how common or uncommon the sizes are for each machine.

Bayes' Theorem is like having a probability calculator that helps us revise existing beliefs based on new evidence. It’s incredibly powerful! Here, it helps us determine the probability that an acceptable cork originated from the first machine.
The theorem is expressed as: \[P(M_1|A) = \frac{P(A|M_1)P(M_1)}{P(A)}\]
Here's what each component means in this context:

• \(P(M_1|A)\) is the probability we want to find: the chance the cork is from the first machine, given it’s acceptable.
• \(P(A|M_1)\) is the calculated probability of the cork being acceptable if it comes from the first machine.
• \(P(M_1)\) is the proportion of total corks produced by the first machine, which is 60% or 0.6.
• \(P(A)\) is the total probability of a cork being acceptable, irrespective of which machine it came from.

By performing the calculation, we discover that the chance of an acceptable cork being from Machine 1 is about 50.6%, displaying how Bayes' Theorem can adjust our understanding based on probabilities.

To determine the overall probability of selecting an acceptable cork from the batch, we factor in both machines' outputs, including how many are considered acceptable.
The probability that a cork is acceptable can be calculated by considering both machines' individual probabilities and their contributions to the total production. This is given by the formula:
\[P(A) = P(A|M_1)P(M_1) + P(A|M_2)P(M_2)\]
Here's the breakdown:

• \(P(A|M_1) = 0.6826\): This represents how many corks from the first machine are acceptable.
• \(P(M_1) = 0.6\): The proportion of corks made by this machine.
• \(P(A|M_2) = 0.9987\): This high number indicates almost all corks from the second machine fall within the acceptable range.
• \(P(M_2) = 0.4\): The portion of corks from the second machine.

By plugging in these values, the total probability is about 0.809, reflecting the combined likelihood of pulling an acceptable cork from either machine. This probability showcase how each machine contributes to meeting acceptable standards and helps complete the probability analysis by blending individual chances together.