Solved on Dec 07, 2023

Design an assembly work table with sitting knee height range between $5^{th}$ percentile women and $95^{th}$ percentile men. Male sitting knee height $\sim \mathcal{N}(21.7\text{ in}, 1.2^2\text{ in}^2)$ , Female $\sim \mathcal{N}(19.2\text{ in}, 1.1^2\text{ in}^2)$ . Find minimum table clearance to fit $95\%$ of men.

STEP 1

Assumptions
1. Male sitting knee heights are normally distributed with a mean ( $\mu_M$ ) of $21.7$ inches.
2. The standard deviation ( $\sigma_M$ ) for male sitting knee heights is $1.2$ inches.
3. Female sitting knee heights are normally distributed with a mean ( $\mu_F$ ) of $19.2$ inches.
4. The standard deviation ( $\sigma_F$ ) for female sitting knee heights is $1.1$ inches.
5. We are looking to find the minimum table clearance for the $95^{\text{th}}$ percentile for men.

STEP 2

To find the $95^{\text{th}}$ percentile of the male sitting knee height distribution, we need to use the standard normal distribution (Z-distribution) to find the Z-score that corresponds to the $95^{\text{th}}$ percentile.

STEP 3

Look up the Z-score that corresponds to the $95^{\text{th}}$ percentile in the standard normal distribution table. Alternatively, use a calculator or software that provides the Z-score for the given percentile.

STEP 4

The Z-score corresponding to the $95^{\text{th}}$ percentile is approximately $1.645$ for a one-tailed test.

STEP 5

Now, we will use the Z-score to find the sitting knee height that corresponds to this percentile for men. The formula to convert a Z-score to an X-value (specific value in a distribution) is:
$X = \mu + Z \cdot \sigma$
Where: - $X$ is the value in the distribution that corresponds to our Z-score. - $\mu$ is the mean of the distribution. - $Z$ is the Z-score corresponding to the desired percentile. - $\sigma$ is the standard deviation of the distribution.

STEP 6

Plug in the values for the mean and standard deviation for men, and the Z-score for the $95^{\text{th}}$ percentile to find the minimum table clearance.
$X_M = \mu_M + Z \cdot \sigma_M$

STEP 7

Calculate the minimum table clearance for men.
$X_M = 21.7\, \text{in.} + 1.645 \cdot 1.2\, \text{in.}$

STEP 8

Perform the multiplication.
$X_M = 21.7\, \text{in.} + 1.645 \cdot 1.2\, \text{in.} = 21.7\, \text{in.} + 1.974\, \text{in.}$

STEP 9

Add the mean to the product of the Z-score and the standard deviation.
$X_M = 21.7\, \text{in.} + 1.974\, \text{in.} = 23.674\, \text{in.}$

STEP 10

Round the result to one decimal place as required.
$X_M \approx 23.7\, \text{in.}$
The minimum table clearance required to satisfy the requirement of fitting $95\%$ of men is approximately $23.7$ inches.

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STEP 1

STEP 2

To find the $95^{\text{th}}$ percentile of the male sitting knee height distribution, we need to use the standard normal distribution (Z-distribution) to find the Z-score that corresponds to the $95^{\text{th}}$ percentile.

STEP 3

Look up the Z-score that corresponds to the $95^{\text{th}}$ percentile in the standard normal distribution table. Alternatively, use a calculator or software that provides the Z-score for the given percentile.

STEP 4

The Z-score corresponding to the $95^{\text{th}}$ percentile is approximately $1.645$ for a one-tailed test.

STEP 5

STEP 6

Plug in the values for the mean and standard deviation for men, and the Z-score for the $95^{\text{th}}$ percentile to find the minimum table clearance.
$X_M = \mu_M + Z \cdot \sigma_M$

STEP 7

Calculate the minimum table clearance for men.
$X_M = 21.7\, \text{in.} + 1.645 \cdot 1.2\, \text{in.}$

STEP 8

Perform the multiplication.
$X_M = 21.7\, \text{in.} + 1.645 \cdot 1.2\, \text{in.} = 21.7\, \text{in.} + 1.974\, \text{in.}$

STEP 9

Add the mean to the product of the Z-score and the standard deviation.
$X_M = 21.7\, \text{in.} + 1.974\, \text{in.} = 23.674\, \text{in.}$

STEP 10

Round the result to one decimal place as required.
$X_M \approx 23.7\, \text{in.}$
The minimum table clearance required to satisfy the requirement of fitting $95\%$ of men is approximately $23.7$ inches.

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STEP 1

STEP 2

To find the 95th95^{\text{th}}95th percentile of the male sitting knee height distribution, we need to use the standard normal distribution (Z-distribution) to find the Z-score that corresponds to the 95th95^{\text{th}}95th percentile.

STEP 3

Look up the Z-score that corresponds to the 95th95^{\text{th}}95th percentile in the standard normal distribution table. Alternatively, use a calculator or software that provides the Z-score for the given percentile.

STEP 4

The Z-score corresponding to the 95th95^{\text{th}}95th percentile is approximately 1.6451.6451.645 for a one-tailed test.

STEP 5

STEP 6

Plug in the values for the mean and standard deviation for men, and the Z-score for the 95th95^{\text{th}}95th percentile to find the minimum table clearance.XM=μM+Z⋅σMX_M = \mu_M + Z \cdot \sigma_MXM​=μM​+Z⋅σM​

STEP 7

Calculate the minimum table clearance for men.XM=21.7 in.+1.645⋅1.2 in.X_M = 21.7\, \text{in.} + 1.645 \cdot 1.2\, \text{in.}XM​=21.7in.+1.645⋅1.2in.

STEP 8

Perform the multiplication.XM=21.7 in.+1.645⋅1.2 in.=21.7 in.+1.974 in.X_M = 21.7\, \text{in.} + 1.645 \cdot 1.2\, \text{in.} = 21.7\, \text{in.} + 1.974\, \text{in.}XM​=21.7in.+1.645⋅1.2in.=21.7in.+1.974in.

STEP 9

Add the mean to the product of the Z-score and the standard deviation.XM=21.7 in.+1.974 in.=23.674 in.X_M = 21.7\, \text{in.} + 1.974\, \text{in.} = 23.674\, \text{in.}XM​=21.7in.+1.974in.=23.674in.

STEP 10

Round the result to one decimal place as required.XM≈23.7 in.X_M \approx 23.7\, \text{in.}XM​≈23.7in.The minimum table clearance required to satisfy the requirement of fitting 95%95\%95% of men is approximately 23.723.723.7 inches.

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To find the $95^{\text{th}}$ percentile of the male sitting knee height distribution, we need to use the standard normal distribution (Z-distribution) to find the Z-score that corresponds to the $95^{\text{th}}$ percentile.

Look up the Z-score that corresponds to the $95^{\text{th}}$ percentile in the standard normal distribution table. Alternatively, use a calculator or software that provides the Z-score for the given percentile.

The Z-score corresponding to the $95^{\text{th}}$ percentile is approximately $1.645$ for a one-tailed test.

Plug in the values for the mean and standard deviation for men, and the Z-score for the $95^{\text{th}}$ percentile to find the minimum table clearance.
$X_M = \mu_M + Z \cdot \sigma_M$

Calculate the minimum table clearance for men.
$X_M = 21.7\, \text{in.} + 1.645 \cdot 1.2\, \text{in.}$

Perform the multiplication.
$X_M = 21.7\, \text{in.} + 1.645 \cdot 1.2\, \text{in.} = 21.7\, \text{in.} + 1.974\, \text{in.}$

Add the mean to the product of the Z-score and the standard deviation.
$X_M = 21.7\, \text{in.} + 1.974\, \text{in.} = 23.674\, \text{in.}$

Round the result to one decimal place as required.
$X_M \approx 23.7\, \text{in.}$
The minimum table clearance required to satisfy the requirement of fitting $95\%$ of men is approximately $23.7$ inches.