Question

in designing a work desk, it is found that males have sitting knee heights with a mean of 22.9 in. and a standard deviation of 1.5 in. (based on data from the department of transportation). use the range rule of thumb to identify (a) the values that are significantly low, (b) the values that are significantly high, and (c) the values that are neither significantly low nor significantly high.

a. what heights are significantly low? select the correct answer below and fill in the answer box(es) to complete your choice. (type integers or decimals. do not round.)
a. heights that are greater than \boxed{} in. and less than \boxed{} in.
b. heights that are greater than \boxed{} in.
c. heights that are less than 19.9 in.

b. what heights are significantly high? select the correct answer below and fill in the answer box(es) to complete your choice. (type integers or decimals. do not round.)
a. heights that are greater than \boxed{} in.
b. heights that are less than \boxed{} in.
c. heights that are greater than \boxed{} in. and less than \boxed{} in.

Explanation:

To solve this problem, we use the range rule of thumb, which states that significant values are typically outside the range of \(\text{mean} - 2 \times \text{standard deviation}\) to \(\text{mean} + 2 \times \text{standard deviation}\). Values within this range are neither significantly low nor high.

Part (a): Significantly Low Heights

Significantly low values are those less than \(\text{mean} - 2 \times \text{standard deviation}\).

Step 1: Calculate the lower bound

Given:

• Mean (\(\mu\)) = 22.9 in
• Standard deviation (\(\sigma\)) = 1.5 in

The formula for the lower bound is:
\[
\text{Lower bound} = \mu - 2\sigma
\]

Substitute the values:
\[
\text{Lower bound} = 22.9 - 2(1.5)
\]

Step 2: Simplify the expression

First, calculate \(2(1.5) = 3\). Then:
\[
\text{Lower bound} = 22.9 - 3 = 19.9
\]

Thus, significantly low heights are less than 19.9 in. The correct option is C. Heights that are less than 19.9 in.

Part (b): Significantly High Heights

Significantly high values are those greater than \(\text{mean} + 2 \times \text{standard deviation}\).

Step 1: Calculate the upper bound

The formula for the upper bound is:
\[
\text{Upper bound} = \mu + 2\sigma
\]

Substitute the values:
\[
\text{Upper bound} = 22.9 + 2(1.5)
\]

Step 2: Simplify the expression

First, calculate \(2(1.5) = 3\). Then:
\[
\text{Upper bound} = 22.9 + 3 = 25.9
\]

Thus, significantly high heights are greater than 25.9 in. The correct option is A. Heights that are greater than 25.9 in.

Part (c): Neither Significantly Low nor High

Values that are neither significantly low nor high lie between the lower and upper bounds:
\[
\text{Lower bound} < \text{Height} < \text{Upper bound}
\]

From Part (a) and (b):

• Lower bound = 19.9 in
• Upper bound = 25.9 in

Thus, heights that are neither significantly low nor high are greater than 19.9 in and less than 25.9 in. The correct option is C. Heights that are greater than 19.9 in. and less than 25.9 in.

Final Answers

a. \(\boldsymbol{\text{C}}\)
b. \(\boldsymbol{\text{A}}\) (with the upper bound as 25.9)
c. \(\boldsymbol{\text{C}}\) (with bounds 19.9 and 25.9)

Answer:

To solve this problem, we use the range rule of thumb, which states that significant values are typically outside the range of \(\text{mean} - 2 \times \text{standard deviation}\) to \(\text{mean} + 2 \times \text{standard deviation}\). Values within this range are neither significantly low nor high.

Part (a): Significantly Low Heights

Significantly low values are those less than \(\text{mean} - 2 \times \text{standard deviation}\).

Step 1: Calculate the lower bound

Given:

• Mean (\(\mu\)) = 22.9 in
• Standard deviation (\(\sigma\)) = 1.5 in

The formula for the lower bound is:
\[
\text{Lower bound} = \mu - 2\sigma
\]

Substitute the values:
\[
\text{Lower bound} = 22.9 - 2(1.5)
\]

Step 2: Simplify the expression

First, calculate \(2(1.5) = 3\). Then:
\[
\text{Lower bound} = 22.9 - 3 = 19.9
\]

Thus, significantly low heights are less than 19.9 in. The correct option is C. Heights that are less than 19.9 in.

Part (b): Significantly High Heights

Significantly high values are those greater than \(\text{mean} + 2 \times \text{standard deviation}\).

Step 1: Calculate the upper bound

The formula for the upper bound is:
\[
\text{Upper bound} = \mu + 2\sigma
\]

Substitute the values:
\[
\text{Upper bound} = 22.9 + 2(1.5)
\]

Step 2: Simplify the expression

First, calculate \(2(1.5) = 3\). Then:
\[
\text{Upper bound} = 22.9 + 3 = 25.9
\]

Thus, significantly high heights are greater than 25.9 in. The correct option is A. Heights that are greater than 25.9 in.

Part (c): Neither Significantly Low nor High

Values that are neither significantly low nor high lie between the lower and upper bounds:
\[
\text{Lower bound} < \text{Height} < \text{Upper bound}
\]

From Part (a) and (b):

• Lower bound = 19.9 in
• Upper bound = 25.9 in

Thus, heights that are neither significantly low nor high are greater than 19.9 in and less than 25.9 in. The correct option is C. Heights that are greater than 19.9 in. and less than 25.9 in.

Final Answers

a. \(\boldsymbol{\text{C}}\)
b. \(\boldsymbol{\text{A}}\) (with the upper bound as 25.9)
c. \(\boldsymbol{\text{C}}\) (with bounds 19.9 and 25.9)