A local kitchen remodel shop works with the middle 74% of the market. The average cost of a kitchen remodel is $22,134, find the minimum and maximum prices that the shop will work with. Assume that the variable is normally distributed and the standard deviation is $3189. Minimum is $ Maximum is $
A local kitchen remodel shop works with the middle 74% of the market. The average cost of a kitchen remodel is $22,134, find the minimum and maximum prices that the shop will work with. Assume that the variable is normally distributed and the standard deviation is $3189. Minimum is $ Maximum is $
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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![**Understanding Kitchen Remodel Costs: Analyzing Market Range**
A local kitchen remodel shop caters to the middle 74% of the market. Given that the average cost of a kitchen remodel is $22,134, their task is to find the minimum and maximum prices for their services. Assume that the cost variable is normally distributed with a standard deviation of $3,189.
To calculate the minimum and maximum prices the shop will work with, let’s consider a few important statistics:
**Battleground of Statistics: Normal Distribution**
A normal distribution is a symmetric, bell-shaped curve characterized by its mean (average) and standard deviation. In this scenario, the mean ($μ$) is $22,134, and the standard deviation ($σ$) is $3,189. The middle 74% of a normal distribution is symbolically represented between z-scores around the mean.
**Finding z-Scores for Middle 74%:**
1. **Understanding the Range**: The middle 74% of a normal distribution leaves 13% in each tail (100% - 74%) / 2.
2. **Using Z-Tables**: For 13% in each tail, the cumulative distribution function (CDF) gives us corresponding z-scores of approximately ±1.07.
**Calculating the Prices:**
1. **Minimum Price Calculation**:
\[
\text{Minimum price} = \mu - (z \times \sigma)
\]
\[
= 22,134 - (1.07 \times 3,189)
\]
\[
\approx 22,134 - 3,413
\]
\[
\approx 18,721
\]
2. **Maximum Price Calculation**:
\[
\text{Maximum price} = \mu + (z \times \sigma)
\]
\[
= 22,134 + (1.07 \times 3,189)
\]
\[
\approx 22,134 + 3,413
\]
\[
\approx 25,547
\]
Therefore, the shop works within the price range of **\$18,721 to \$25,547**.
**Answer Fields:**
- **Minimum is \$ [enter calculated value]**
- **Maximum is \$ [enter calculated value]**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb845ac35-162e-4da5-a6d5-ce13a46079d9%2Fa3e9eb38-5816-46e1-9b57-1c60f9a10185%2Fcmcl1v3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Kitchen Remodel Costs: Analyzing Market Range**
A local kitchen remodel shop caters to the middle 74% of the market. Given that the average cost of a kitchen remodel is $22,134, their task is to find the minimum and maximum prices for their services. Assume that the cost variable is normally distributed with a standard deviation of $3,189.
To calculate the minimum and maximum prices the shop will work with, let’s consider a few important statistics:
**Battleground of Statistics: Normal Distribution**
A normal distribution is a symmetric, bell-shaped curve characterized by its mean (average) and standard deviation. In this scenario, the mean ($μ$) is $22,134, and the standard deviation ($σ$) is $3,189. The middle 74% of a normal distribution is symbolically represented between z-scores around the mean.
**Finding z-Scores for Middle 74%:**
1. **Understanding the Range**: The middle 74% of a normal distribution leaves 13% in each tail (100% - 74%) / 2.
2. **Using Z-Tables**: For 13% in each tail, the cumulative distribution function (CDF) gives us corresponding z-scores of approximately ±1.07.
**Calculating the Prices:**
1. **Minimum Price Calculation**:
\[
\text{Minimum price} = \mu - (z \times \sigma)
\]
\[
= 22,134 - (1.07 \times 3,189)
\]
\[
\approx 22,134 - 3,413
\]
\[
\approx 18,721
\]
2. **Maximum Price Calculation**:
\[
\text{Maximum price} = \mu + (z \times \sigma)
\]
\[
= 22,134 + (1.07 \times 3,189)
\]
\[
\approx 22,134 + 3,413
\]
\[
\approx 25,547
\]
Therefore, the shop works within the price range of **\$18,721 to \$25,547**.
**Answer Fields:**
- **Minimum is \$ [enter calculated value]**
- **Maximum is \$ [enter calculated value]**
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