Question

Quality Control A company manufactures wooden yardsticks. The lengths of the yardsticks are normally distributed with a mean of 36 inches and a standard deviation of \(0.2\) inch. Find the probability that a yardstick is (a) longer than \(35.5\) inches. (b) longer than \(35.9\) inches.

Short answer

(a) The probability that a yardstick is longer than 35.5 inches is approximately 0.9938. (b) The probability that a yardstick is longer than 35.9 inches is approximately 0.6915.

Step-by-step solution

Calculate Z-Score for 35.5 inches

Calculate the z-score for 35.5 inches using the formula \(z = (x - \mu) / \sigma\). Therefore, \(z = (35.5 - 36) / 0.2 = -2.5\).

Find Probability for Z-Score

We look up the z-score in a z-table, which gives us the probability that a yardstick is less than 35.5 inches. The value we get from the table is 0.0062. However, we want the probability that the yardstick is longer than 35.5 inches, so we do 1 - 0.0062 = 0.9938.

Calculate Z-Score for 35.9 inches

Calculate the z-score for 35.9 inches using the formula \(z = (x - \mu) / \sigma\). Therefore, \(z = (35.9 - 36) / 0.2 = -0.5\).

Find Probability for Z-Score

We look up the z-score in a z-table, which gives us the probability that a yardstick is less than 35.9 inches. The value we get from the table is 0.3085. However, we want the probability that the yardstick is longer than 35.9 inches, so we do 1 - 0.3085 = 0.6915.

Key concepts

Quality Control

When a company is manufacturing products, like wooden yardsticks, quality control is crucial to ensure each product meets specific standards. Quality control often uses statistical methods to verify that a product's features, such as length, fall within defined limits. These methods are particularly useful in identifying products that deviate from the desired measurements and determining how frequently these deviations occur.

By utilizing the properties of the normal distribution, a company can assess whether a batch of products, such as yardsticks, meets the quality standards. By analyzing the mean and standard deviation, companies can predict the occurrence of defects, allowing them to improve the manufacturing process if necessary and maintain high quality.

Mean and Standard Deviation

In a normal distribution, the **mean** is the point of symmetry for the data, indicating the average value. For wooden yardsticks that are manufactured to be 36 inches long, the mean is 36 inches. This mean represents the average length that most of the yardsticks should have.

The **standard deviation** measures the variability or spread of the data. In this scenario, with a standard deviation of 0.2 inches, it indicates how much the length of a yardstick may differ from the mean. The smaller the standard deviation, the less variation in the lengths of the yardsticks — a vital factor in maintaining consistent quality.

• If most data points cluster closely to the mean, it means production is in good control.
• Larger variations could signal problems in the manufacturing process that need addressing.

Z-Score Calculation

Z-scores are a way to determine how far away a data point is from the mean, measured in standard deviations. Calculating a z-score helps compare different data points and determine probabilities. The z-score formula is: \[z = \frac{x - \mu}{\sigma}\]where:

• \(x\) is the data point (e.g., a yardstick's length),
• \(\mu\) is the mean,
• \(\sigma\) is the standard deviation.

In our exercise, the z-scores were calculated for yardsticks longer than 35.5 inches and 35.9 inches. These z-scores tell us how these measurements compare to the average length. Once calculated, these scores are looked up on a z-table to find probabilities.
If a z-score is negative, the measurement is below the mean. Conversely, a positive z-score indicates the measurement is above the mean. Understanding z-scores is instrumental for assessing how unusual or typical a particular measurement is in a given distribution.