A contractor has concluded from his experience that the cost...

A contractor has concluded from his experience that the cost of building a luxury home is a normally distributed random variable with a mean of $\$ 500,000$ and a standard deviation of $\$ 50,000$. a. What is the probability that the cost of building a home will be between $\$ 460,000$ and $\$ 540,000$ ? b. The probability is $0.2$ that the cost of building will be less than what amount? c. Find the shortest range such that the probability is $0.95$ that the cost of a luxury home will fall in this range.

A contractor has concluded from his experience that the cost of building a luxury home is a normally distributed random variable with a mean of $\$ 500,000$ and a standard deviation of $\$ 50,000$.
a. What is the probability that the cost of building a home will be between $\$ 460,000$ and $\$ 540,000$ ?
b. The probability is $0.2$ that the cost of building will be less than what amount?
c. Find the shortest range such that the probability is $0.95$ that the cost of a luxury home will fall in this range.

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean. It describes how the values of a variable are distributed with most of the observations clustering around the mean and probabilities tapering off equally on both sides. Its bell shape is characterized by properties that make it useful for modeling many natural phenomena due to the Central Limit Theorem.

Mean and Standard Deviation

The mean is the central value representing the average of the data, while the standard deviation measures the spread or dispersion around the mean. In the context of a normal distribution, these parameters define the location and the scale, allowing us to understand and characterize the distribution's behavior.

Standardization (Z-Scores)

Standardization is the process of converting a value from a normal distribution into a z-score by subtracting the mean and dividing by the standard deviation. This conversion allows us to compare different normal distributions and assess probabilities using the standard normal distribution (with a mean of 0 and a standard deviation of 1).

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a continuous random variable gives the probability that the variable will be less than or equal to a particular value. It is a key tool when calculating the probability of a value falling within a given range and is essential for tasks like finding the area under the curve between two points.

Quantile (Inverse CDF) Calculation

Quantile calculation, or using the inverse CDF, involves finding the value below which a given percentage of the data in the distribution falls. This is commonly used to determine thresholds, such as the point below which there is a 20% chance of occurrence, facilitating decisions based on percentile ranks.

Confidence Interval for Normal Distribution

A confidence interval in the context of a normal distribution specifies a range around the mean within which a certain percentage of the data is expected to fall. The shortest such interval for a given probability level is symmetric around the mean and is calculated using the quantiles from the standard normal distribution, which aids in understanding the variability and reliability of estimates.

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