Suppose that you are designing an instrument panel for a large industrial machine. The machine requires the person using it to reach 2 feet from a particular position. The reach from this position for adult women is known to have a mean of 2.8 feet with a standard deviation of .5. The reach for adult men is known to have a mean of 3.1 feet with a standard deviation of .6. Both women’s and men’s reach from this position is normally distributed. If this design is implemented:
What percentage of women will not be able to work on this instrument panel?
What percentage of men will not be able to work on this instrument panel?
for women, 2 ft is 1.6 std below the mean
for men, 2 ft is 3.33 std below the mean.
So, look those up in your Z table to see the break points.
Or, play around at
http://davidmlane.com/hyperstat/z_table.html
.For women:
Mean = 2.8
SD = 0.5
P( x < 2) =
μ = 2.8
σ = 0.5
standardize x to z = (x - μ) / σ
P(x < 2) = P( z < (2-2.8) / 0.5)
= P(z < -1.6) = 0.0548 ---- 5.48 percent of women won't be able to work on this instrument panel
(From Normal probability table)
Men:
Mean = 3.1
SD = 0.6
P( x < 2) =
μ = 3.1
σ = 0.6
standardize x to z = (x - μ) / σ
P(x < 2) = P( z < (2-3.1) / 0.6)
= P(z < -1.8333) = 0.0336 --- 3.36 percent of men won't be able to work on this instrument panel
(From Normal probability table)
To find the percentage of women who will not be able to work on the instrument panel, we need to calculate the probability of a woman's reach being less than 2 feet.
Step 1: Standardize the value of 2 feet using the z-score formula: z = (x - μ) / σ
z = (2 - 2.8) / 0.5
= -0.8 / 0.5
= -1.6
Step 2: Look up the z-score in the standard normal distribution table. The area to the left of -1.6 is approximately 0.0548.
Step 3: Convert the decimal to a percentage by multiplying by 100: 0.0548 * 100 = 5.48%
Therefore, approximately 5.48% of women will not be able to work on this instrument panel.
To find the percentage of men who will not be able to work on the instrument panel, we follow the same steps:
Step 1: Standardize the value of 2 feet using the z-score formula: z = (x - μ) / σ
z = (2 - 3.1) / 0.6
= -1.9 / 0.6
= -3.16
Step 2: Look up the z-score in the standard normal distribution table. The area to the left of -3.16 is approximately 0.0008.
Step 3: Convert the decimal to a percentage by multiplying by 100: 0.0008 * 100 = 0.08%
Therefore, approximately 0.08% of men will not be able to work on this instrument panel.
To determine the percentage of women who will not be able to work on the instrument panel, we can use the concept of z-scores.
Step 1: Find the z-score for the given reach value for women using the formula: z = (x - μ) / σ
where x is the desired reach value, μ is the mean reach value, and σ is the standard deviation.
Given:
x (desired reach) = 2 feet
μ (mean reach for women) = 2.8 feet
σ (standard deviation for women) = 0.5 feet
z = (2 - 2.8) / 0.5 = -1.6
Step 2: Find the corresponding percentage using the z-score table or a calculator. The z-score table shows the area under the normal distribution curve up to a specific z-score. In this case, we want to find the area to the left of the z-score of -1.6.
Looking up the z-score of -1.6 in the z-score table, we find that the corresponding area is approximately 0.0548 or 5.48%.
Therefore, approximately 5.48% of women will not be able to work on this instrument panel.
To find the percentage of men who will not be able to work on the instrument panel, we follow the same steps.
Step 1: Find the z-score for the given reach value for men.
Given:
x (desired reach) = 2 feet
μ (mean reach for men) = 3.1 feet
σ (standard deviation for men) = 0.6 feet
z = (2 - 3.1) / 0.6 = -1.833
Step 2: Find the corresponding percentage using the z-score table.
The z-score of -1.833 corresponds to an area of approximately 0.0336 or 3.36%.
Therefore, approximately 3.36% of men will not be able to work on this instrument panel.