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------------------------------------------------ Assume that weights are normally distributed with a mean of [tex]\mu = 143[/tex] lb and a standard deviation of [tex]\sigma = 29[/tex] lb. For a single part, find the probability that the weight is above 177 lb.

To find the probability of a weight above 177 lb with a mean of 143 lb and a standard deviation of 29 lb, calculate the z-score and then use the z-score to determine the right-tail probability from the standard normal distribution.

Explanation:

To find the probability that the weight is above 177 lb when the weights are normally distributed with a mean (μ) of 143 lb and a standard deviation (σ) of 29 lb, we first calculate the z-score for 177 lb. The z-score is given by:

Z = (X - μ) / σ

Where X is the value we're interested in, which is 177 lb.

Z = (177 - 143) / 29
Z = 34 / 29
Z ≈ 1.1724

Now that we have the z-score, we use the standard normal distribution table, or a calculator with normal distribution capabilities, to find the probability that a single part weighs more than 177 lb, which corresponds to P(Z > 1.1724).

This gives us a probability that is usually the area to the right of the z-score on the standard normal curve. Since standard normal distribution tables typically provide the area to the left, we calculate:

P(Z > 1.1724) = 1 - P(Z < 1.1724)

Using a standard normal distribution table or calculator will give us the desired probability.