Final answer:
To find the cut point values for the middle 40% of steer weights, identify the z-scores for the 30th and 70th percentiles and apply them to the formula X = (z * σ) + μ to convert them to weights. The cut points are approximately 1120.32 pounds (30th percentile) and 1183.68 pounds (70th percentile) for a normal distribution with a mean of 1152 and a standard deviation of 84.
Step-by-step explanation:
To find the cut point values for the middle 40% of steer weights based on the model n(1152,84), which is a normal distribution with a mean (μ) of 1152 pounds and a standard deviation (σ) of 84 pounds, we need to determine the z-scores that correspond to the 30th percentile and the 70th percentile. Since the middle 40% of the data lies between these two percentiles.
Step-by-Step Process:
- First, consult a standard normal distribution table (z-table) or use a statistical software/calculator to find the z-scores corresponding to the 30th and 70th percentiles. A z-table may not always provide exact values for these percentiles, so software or a calculator is often the preferable tool for accuracy.
- For the 30th percentile, the z-score is approximately -0.52, and for the 70th percentile, it is approximately 0.52. These values may vary slightly depending on the source, but they are typically close to these values.
- Use the z-score formula:
Z = (X - μ) / σ, where X is the cut point weight we are solving for, to convert the z-scores back to weights (X) in the context of the given normal distribution. - For the 30th percentile: X = (z * σ) + μ, therefore X = (-0.52 * 84) + 1152, which calculates to approximately 1120.32 pounds.
- For the 70th percentile: X = (z * σ) + μ, so X = (0.52 * 84) + 1152, which calculates to approximately 1183.68 pounds.
The cut point values for the middle 40% of the weights are approximately 1120.32 pounds for the lower cut and 1183.68 pounds for the upper cut.