Solved on Nov 07, 2023
Find the proportion of steel rods with length less than , given the rods have a mean length of and standard deviation of .
STEP 1
Assumptions1. The mean length of the rods is25 cm. The standard deviation is0.07 cm3. The lengths of the rods are normally distributed4. We are looking for the proportion of rods with a length less than24.9 cm
STEP 2
We can use the Z-score formula to standardize the length of24.9 cm. The Z-score is a measure of how many standard deviations an element is from the mean. The formula for the Z-score iswhere- X is the value we are standardizing (24.9 cm in this case) - is the mean (25 cm) - is the standard deviation (0.07 cm)
STEP 3
Substitute the given values into the Z-score formula
STEP 4
Calculate the Z-score
STEP 5
The Z-score tells us that24.9 cm is approximately1.43 standard deviations below the mean. To find the proportion of rods with a length less than24.9 cm, we need to find the area to the left of -1.43 on the standard normal distribution. This is done by looking up the Z-score in a standard normal distribution table or using a calculator with a normal distribution function.
STEP 6
Using a standard normal distribution table or a calculator with a normal distribution function, we find that the area to the left of -1.43 is approximately0.0764.
Therefore, the proportion of rods with a length less than24.9 cm is0.0764, or.64% when expressed as a percentage.
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