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A total of 100 hardness measurements are performed on a large slab of steel. If, using the Rockwell C-scale, the mean of the measurement is 60 and the standard deviation is 7.5. a) How many of the measurements can be expected to fall between the hardness reading of 55 and 65 Rockwell C-scale? b) How many of measurements falls under 75% of the populations. c) How many of the measurements can be expected to fall between the hardness reading of 40 and 70 Rockwell C-scale?

User Diralik
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Answer:

To answer these questions, you can use the properties of the normal distribution since you have the mean (μ) and standard deviation (σ) of the hardness measurements. The Rockwell C-scale measurements can be approximated as normally distributed with μ = 60 and σ = 7.5.

a) How many of the measurements can be expected to fall between the hardness readings of 55 and 65 on the Rockwell C-scale?

To find the proportion of measurements between 55 and 65, you can use the z-score formula:

\[Z = \frac{X - \mu}{\sigma}\]

Where:

- X is the value you want to find the proportion for (55 and 65 in this case).

- μ is the mean (60).

- σ is the standard deviation (7.5).

For X = 55:

\[Z_1 = \frac{55 - 60}{7.5} = -0.67\]

For X = 65:

\[Z_2 = \frac{65 - 60}{7.5} = 0.67\]

Now, you can use a standard normal distribution table or calculator to find the proportion between these z-scores:

\[P(-0.67 < Z < 0.67)\]

You can find this probability from a standard normal distribution table or use a calculator, which is approximately 0.5040.

To find the number of measurements falling in this range, multiply this probability by the total number of measurements (100):

Number of measurements = 0.5040 * 100 = 50.4

Since you cannot have a fraction of a measurement, you can expect approximately 50 measurements to fall between 55 and 65 on the Rockwell C-scale.

b) How many of the measurements fall under 75% of the population?

To find the value below which 75% of the measurements fall, you can use the z-score corresponding to the 75th percentile (0.75). You can either use a standard normal distribution table or a calculator to find this value.

For the 75th percentile:

\[Z = 0.67\]

Now, convert this z-score back to the actual hardness measurement:

\[X = Z * σ + μ = 0.67 * 7.5 + 60 = 65.025\]

So, 75% of the measurements fall below a hardness reading of approximately 65.03 on the Rockwell C-scale.

c) How many of the measurements can be expected to fall between the hardness readings of 40 and 70 on the Rockwell C-scale?

First, find the z-scores for X = 40 and X = 70 using the same formula:

For X = 40:

\[Z_1 = \frac{40 - 60}{7.5} = -2.67\]

For X = 70:

\[Z_2 = \frac{70 - 60}{7.5} = 1.33\]

Now, find the proportion between these z-scores:

\[P(-2.67 < Z < 1.33)\]

You can find this probability from a standard normal distribution table or use a calculator, which is approximately 0.9082.

To find the number of measurements falling in this range, multiply this probability by the total number of measurements (100):

Number of measurements = 0.9082 * 100 = 90.82

Since you cannot have a fraction of a measurement, you can expect approximately 91 measurements to fall between 40 and 70 on the Rockwell C-scale.

Explanation:

User Mike Muller
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