A seafood shop sells salmon fillets where the weight of each fillet is normally distributed with a mean of 1.6 pounds and a standard deviation of 0.3 pounds. They want to classify the largest fillets as extra large and charge a higher price for them. If they want the largest 15 percent of the fillets to be classified as extra large, the minimum weight for an extra large fillet should be 1.91 pounds.

Question

Quizplus · Accepted Answer

To find the minimum weight that classifies the largest 15% of the salmon fillets as extra large, we use the Z-score table or a calculator for the normal distribution. The Z-score corresponding to the top 15% (or the 85th percentile, since the normal distribution is symmetric) is approximately 1.036. Using the Z-score formula $Z = \frac{X - \mu}{\sigma}$, where $X$ is the value we seek, $\mu = 1.6$ pounds is the mean, and $\sigma = 0.3$ pounds is the standard deviation, we solve for $X$: $1.036 = \frac{X - 1.6}{0.3}$. Solving this equation gives $X \approx 1.91$ pounds, which matches the statement.

A Seafood Shop Sells Salmon Fillets Where the Weight of Each