A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 205.6-cm and a standard deviation of 1.1-cm. For shipment, 10 steel rods are bundled together. Find P25, which is the average length separating the smallest 25% bundles from the largest 75% bundles. P25

P25 = 205.365cm

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central limit theorem:

The Central Limit Theorem estabilishes that, for a random variable X, with mean and standard deviation , the sample means with size n can be approximated to a normal distribution with mean and standard deviation

In this problem, we have that:

Find P25, which is the average length separating the smallest 25% bundles from the largest 75% bundles. P25

This is the value of X when Z has a pvalue of 0.25. So X when Z = -0.675.

So

By the Central Limit Theorem

P25 = 205.365cm

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step-by-step explanation: