A manufacturer of electroluminescent lamps knows that the...

A manufacturer of electroluminescent lamps knows that the amount of luminescent ink deposited on one of its products is normally distributed with a mean of 1.2 grams and a standard deviation of 0.03 gram. Any lamp with less than 1.14 grams of luminescent ink will fail to meet customers' specifications. A random sample of 25 lamps is collected and the mass of luminescent ink on each is measured. (a) What is the probability that at least one lamp fails to meet specifications? (b) What is the probability that five lamps or fewer fail to meet specifications? (c) What is the probability that all lamps conform to specifications? (d) Why is the joint probability distribution of the 25 lamps not needed to answer the previous questions?

A manufacturer of electroluminescent lamps knows that the amount of luminescent ink deposited on one of its products is normally distributed with a mean of 1.2 grams and a standard deviation of 0.03 gram. Any lamp with less than 1.14 grams of luminescent ink will fail to meet customers' specifications. A random sample of 25 lamps is collected and the mass of luminescent ink on each is measured.
(a) What is the probability that at least one lamp fails to meet specifications?
(b) What is the probability that five lamps or fewer fail to meet specifications?
(c) What is the probability that all lamps conform to specifications?
(d) Why is the joint probability distribution of the 25 lamps not needed to answer the previous questions?

Key Concepts

Joint Probability Distribution

A joint probability distribution would theoretically describe the probabilities of outcomes for multiple variables simultaneously. However, because each lamp is produced independently, the outcome for each one can be considered separately. This independence negates the need for a complex joint distribution, as the probability for a set of lamps can be computed as the product of the individual probabilities.

Independence of Measurements

In the context of quality control, each product is assumed to be produced independently. This means that the measurement (or outcome) for one lamp does not influence another. This independence allows us to apply the binomial model and simpler probability calculations without needing to account for correlations between measurements.

Binomial Distribution

This distribution models the number of successes in a fixed number of independent trials, where each trial has only two outcomes (such as pass or fail). When assessing a sample of lamps, if each lamp independently meets or fails specifications based on a known probability, the binomial distribution is used to calculate the probabilities of a specific number of failures (or conforming lanterns) in the sample.

Z-Score and Standardization

The Z-score is a statistical metric that indicates how many standard deviations an observation is away from the mean. It is used to standardize different datasets or observations and to calculate probabilities associated with specific values under a normal distribution. In quality control, converting measurements to Z-scores helps determine the proportion of items that fall below or above a threshold value.

Normal Distribution

This concept describes a continuous probability distribution where real-valued random variables are symmetrically distributed about the mean. In quality control and manufacturing, the normal distribution models variations in product characteristics, such as the amount of luminescent ink deposited. It is characterized by its mean and standard deviation, which together determine the likelihood of observations falling within any given range.

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