Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Nov 07, 2023

Find the proportion of steel rods with length less than $24.9 \mathrm{cm}$ , given the rods have a mean length of $25 \mathrm{cm}$ and standard deviation of $0.07 \mathrm{cm}$ .

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 is $Z = \frac{X - \mu}{\sigma}$ where- X is the value we are standardizing (24.9 cm in this case) - $\mu$ is the mean (25 cm) - $\sigma$ is the standard deviation (0.07 cm)

STEP 3

Substitute the given values into the Z-score formula $Z = \frac{24.9 -25}{0.07}$

STEP 4

Calculate the Z-score $Z = \frac{24.9 -25}{0.07} = -1.42857$

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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