Question

23. The mean weight of trucks traveling on a particular section of 1-475 is not known. A state highway inspector needs an estimate of the mean. He selects a random sample of 49 trucks passing the weighing station and finds the mean is 15.8 tons, with a standard deviation of the sample of 4.2 tons. What is probability that a truck will weigh less than 14.3 tons

Answer 1

0.3594 = 35.94% probability that a truck will weigh less than 14.3 tons

Explanation:

To solve this question, we need to understand the normal probability distribution.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

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 p-value, we get the probability that the value of the measure is greater than X.

Mean is 15.8 tons, with a standard deviation of the sample of 4.2 tons.

This means that
`\mu = 15.8, \sigma = 4.2`

What is probability that a truck will weigh less than 14.3 tons?

This is the pvalue of Z when X = 14.3. So

`Z = (X - \mu)/(\sigma)`

`Z = (14.3 - 15.8)/(4.2)`

`Z = -0.36`

`Z = -0.36` has a pvalue of 0.3594

0.3594 = 35.94% probability that a truck will weigh less than 14.3 tons