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: