Synthetic fiber strings from a supplier are known to have tensile strengths that are normally distributed with a mean of 520 kilo Newtons per square meter and a standard deviation of 30 kilo Newtons per square meter.

What is the probability that a randomly selected synthetic fiber string from the supplier has a tensile strength between 480 kilo Newtons per square meter and 510 kilo Newtons per square meter?

The probability that a randomly selected synthetic fibre string from the supplier has tensile strength between 480 kN/m and 510 kN/m is approximately 0.2107. To calculate this probability, we need to standardize the values and use the z-scores to find the area under the standard normal distribution curve using a z-table or a calculator.

Explanation:

The question is asking for the probability that a randomly selected synthetic fibre string from the supplier has a tensile strength between 480 kilo Newtons per square metre and 510 kilo Newtons per square metre. To solve this, we need to standardize the values and calculate the z-scores. Then, we can use these z-scores to find the area under the standard normal distribution curve using a z-table or a calculator.

First, let's compute the z-score for a tensile strength of 480 kN/m:
z = (x - mean) / standard deviation = (480 - 520) / 30 = -1

Next, let's compute the z-score for a tensile strength of 510 kN/m:
z = (x - mean) / standard deviation = (510 - 520) / 30 = -0.333

Now, we can use the z-scores to find the area under the standard normal distribution curve. From a z-table, we can find that the area to the left of -1 is 0.1587 and the area to the left of -0.333 is 0.3694. To find the probability between these two z-scores, we subtract the smaller probability from the larger probability:
P(480 < x < 510) = 0.3694 - 0.1587 = 0.2107

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