7. a factory has a machine that makes steel rods. The length of the rods that the machine makes follows a normal distribution curve with a mean of 11.5 feet and a standard deviation of 0.3 feet. The factory manager does not want to use the rods from the bottom 20% and the top 15%. Find the rod length the manager will be using as the basis for separating the rods.

Question

7. a factory has a machine that makes steel rods. The length of the rods that the machine makes follows a normal distribution curve with a mean of 11.5 feet and a standard deviation of 0.3 feet. The factory manager does not want to use the rods from the bottom 20% and the top 15%. Find the rod length the manager will be using as the basis for separating the rods.

a). 11.56 and 11.86
b). 11.55 and 11.74
c). 11.19 and 11.75
d). 11.25 and 11.81

Answer 1

use a z-score table to find the lengths represented by bottom 20% and the top 15% of the population

the top is ≈ 1.04 s.d. above the mean

the bottom is ≈ 0.84 s.d. below the mean

Answer 2

Z = (score-mean)/SD

1.04 = (score-11.5)/.3

Solve for top score, repeat process for bottom score.

Answer 3

To find the rod length that the manager will be using as the basis for separating the rods, we need to determine the cutoff values for the bottom 20% and top 15% of the distribution.

Step 1: Find the z-scores corresponding to the percentiles.

To find the z-score for the bottom 20%, we use the formula:
z-score = invNorm(percentage/100)
For the bottom 20%, the percentage is 20, so the z-score is:
z1 = invNorm(20/100) = invNorm(0.2)

To find the z-score for the top 15%, we use the formula:
z2 = invNorm(85/100) = invNorm(0.85)

Step 2: Convert the z-scores to rod lengths.

To convert z-scores to rod lengths, we use the formula:
rod length = mean + (z-score * standard deviation)

For the bottom 20%:
rod length1 = 11.5 + (z1 * 0.3)

For the top 15%:
rod length2 = 11.5 + (z2 * 0.3)

Now, we can calculate the rod lengths:

rod length1 = 11.5 + (invNorm(0.2) * 0.3)
rod length1 ≈ 11.5 + (-0.841621 * 0.3)
rod length1 ≈ 11.5 - 0.2524863
rod length1 ≈ 11.2475137

rod length2 = 11.5 + (invNorm(0.85) * 0.3)
rod length2 ≈ 11.5 + (1.03643 * 0.3)
rod length2 ≈ 11.5 + 0.310929
rod length2 ≈ 11.810929

Therefore, the rod lengths the manager will be using as the basis for separating the rods are approximately 11.25 and 11.81 feet.

Answer 4

To find the rod length the factory manager will be using as the basis for separating the rods, we need to find the cutoff values for the bottom 20% and the top 15% of the rod lengths.

First, we need to find the z-scores corresponding to the cutoff values.

For the bottom 20%, we need to find the z-score that represents the 20th percentile. The formula to find the z-score corresponding to a given percentile is:

z = (x - mean) / standard deviation

Let's find the z-score for the bottom 20%:

z_20 = invNorm(0.2) (using a standard normal distribution table or calculator)

Next, we need to calculate the rod length corresponding to this z-score using the formula:

x_20 = mean + z_20 * standard deviation

For the top 15%, we need to find the z-score that represents the 85th percentile (1 - 0.15 = 0.85). Let's find the z-score for the top 15%:

z_85 = invNorm(0.85)

Then, calculate the rod length corresponding to this z-score:

x_85 = mean + z_85 * standard deviation

Finally, we can determine the rod length the manager will be using as the basis for separating the rods by rounding the calculated values to two decimal places:

The options are:
a). 11.56 and 11.86
b). 11.55 and 11.74
c). 11.19 and 11.75
d). 11.25 and 11.81

By calculating the values for x_20 and x_85, we can compare them to the provided options:

x_20 ≈ 11.19 and x_85 ≈ 11.75

Therefore, the correct answer is c). 11.19 and 11.75.