Question

The distribution of the number of items produced by an assembly line during an 8 -hour shift can be approximated by a normal distribution with mean value 150 and standard deviation 10 . a. What is the probability that the number of items produced is at most \(120 ?\) b. What is the probability that at least 125 items are produced? c. What is the probability that between 135 and 160 (inclusive) items are produced?

Short answer

The probabilities are as follows: (a) A number less than or equal to the z-score value corresponding to 120. (b) 1 minus the z-score value corresponding to 125, because at least 125 products means all probabilities excluding less than 125. (c) The z-score of 160 minus the z-score of 135, because between 135 and 160 includes all probabilities within this range.

Step-by-step solution

Calculate the Z Scores for given numbers

A Z score represents how many standard deviations an element is from the mean. You calculate it by using the formula Z = (X - µ)/σ, where X is the given number. For part a, replace X with 120, for part b, replace X with 125 and for part c, replace X with 135 and 160 respectively.

Interpret and Use the Z Scores

Once the Z Scores have been calculated, apply them and use a Z-Table to find probabilities. For part a, check the value for the calculated Z in the Z-table to get the probability directly. For part b, since it is asking for the number of items 'at least' produced it would be 1-P where P is the value got from the Z table. For part c, find the probability for both Z(135) and Z(160) from the Z-table and subtract Z(135) from Z(160).

Present the Findings

Using the tabulated probabilities, present the answers to: (a) the probability that the number of items produced is at most 120. (b) the probability that at least 125 items are produced. (c) the probability that between 135 and 160 items are produced.

Key concepts

Z Score Calculation

Understanding the Z score is crucial when working with normal distributions, as it allows us to determine how far away a particular value is from the mean, in terms of standard deviations. The Z score calculation is a standard procedure to standardize an individual data point relative to the mean and standard deviation of the dataset.

The formula for calculating a Z score is:\[ Z = \frac{(X - \mu)}{\sigma} \] where:\[ \begin{align*} X &\text{ is the value in question,}\ \mu &\text{ is the mean of the dataset, and}\ \sigma &\text{ is the standard deviation of the dataset.} \end{align*} \]
In our exercise, the assembly line's production numbers are normally distributed with a mean (\(\mu\)) of 150 items and a standard deviation (\(\sigma\)) of 10 items. For example, to calculate the Z score for producing 120 items, we subtract the mean from 120 and divide by the standard deviation, which illustrates how many standard deviations below the mean 120 items are.

Standard Deviation

Standard deviation (often denoted as \(\sigma\)) is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values are close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

In real-world terms, considering the assembly line example, a standard deviation of 10 implies that the production numbers typically vary by 10 items from the mean (150) on an 8-hour shift. The concept of standard deviation is fundamental in calculating Z scores and assessing probabilities because it forms the denominator of the Z score formula, effectively scaling the difference between a specific value and the mean. Thus, the standard deviation directly influences the Z score magnitude and, by extension, the probability outcomes.

Z-table Application

After calculating Z scores, the next step is to apply these values using a Z-table, which lists the probabilities of a standard normal distribution. A Z-table reflects the probability of a random variable falling below a specified Z score in a standard normal distribution.

Here is how to use the Z-table in our exercise scenarios:

• For part (a), we look up the Z score for the value 120 and find the associated cumulative probability in the Z-table. This probability represents the chance that on a random 8-hour shift, the assembly line will produce at most 120 items.
• For part (b), we'd analyze the Z score for 125 items. The Z-table gives us the probability for being below this value, but we want 'at least' 125 items, so we subtract the table value from 1 to get the probability of producing 125 or more items.
• In part (c), we're interested in the range between 135 and 160 items. We find the cumulative probabilities for both Z scores, then subtract the lower probability (Z for 135) from the higher probability (Z for 160) to obtain the chance of producing between 135 and 160 items.

By understanding Z-table application, students can translate Z scores into meaningful probabilities and gain insights into the data's behavior in relation to the entire distribution.