Problem 5 - Basic Statistics for Business and Economics

Question

The April rainfall in Flagstaff, Arizona, follows a uniform distribution between 0.5 and 3.00 inches. a. What are the values for \(a\) and \(b\) ? b. What is the mean amount of rainfall for the month? What is the standard deviation? c. What is the probability of less than an inch of rain for the month? d. What is the probability of "exactly" 1.00 inch of rain? e. What is the probability of more than 1.50 inches of rain for the month?

Step-by-Step Solution

Answer

a. \(a = 0.5\), \(b = 3.0\); b. Mean = 1.75, SD = 0.7217; c. 0.2; d. 0; e. 0.6.

Step 1: Identify the Uniform Distribution Parameters

In a uniform distribution, the values for \(a\) and \(b\) represent the lower and upper bounds of the distribution. For April rainfall in Flagstaff, Arizona, \(a = 0.5\) inches and \(b = 3.00\) inches.

Step 2: Calculate the Mean of the Distribution

The mean \( \mu \) of a uniform distribution is given by the formula \( \mu = \frac{a+b}{2} \). Substitute \(a = 0.5\) and \(b = 3.0\) to find the mean: \[ \mu = \frac{0.5 + 3.0}{2} = 1.75 \text{ inches}. \]

Step 3: Calculate the Standard Deviation of the Distribution

The standard deviation \( \sigma \) of a uniform distribution is calculated with the formula \( \sigma = \frac{b-a}{\sqrt{12}} \). Using \(a = 0.5\) and \(b = 3.0\), we find: \[ \sigma = \frac{3.0 - 0.5}{\sqrt{12}} \approx 0.7217 \text{ inches}. \]

Step 4: Calculate the Probability of Less Than 1 Inch of Rain

The probability of \(X\) being less than a certain value in a continuous uniform distribution is given by \( P(X < x) = \frac{x-a}{b-a} \). For \(x = 1.0\), \[ P(X < 1.0) = \frac{1.0 - 0.5}{3.0 - 0.5} = \frac{0.5}{2.5} = 0.2. \]

Step 5: Determine the Probability of Exactly 1 Inch of Rain

In a continuous uniform distribution, the probability of an exact value is always 0. Therefore, the probability of exactly 1.00 inch of rain is: \[ P(X = 1.0) = 0. \]

Step 6: Calculate the Probability of More Than 1.5 Inches of Rain

For \(x > 1.5\), the probability is \( P(X > x) = \frac{b-x}{b-a} \). Using \(x = 1.5\), calculate: \[ P(X > 1.5) = \frac{3.0 - 1.5}{3.0 - 0.5} = \frac{1.5}{2.5} = 0.6. \]

Key Concepts

Mean Calculation

To find the mean of a uniform distribution where the data is evenly spread between two numbers, we use a simple and intuitive formula. The mean is essentially the middle point between the smallest and largest value of the distribution.

Here's how you calculate it: Take the sum of the lower bound (a) and the upper bound (b) of the distribution and then divide by 2.

It's expressed as: \[ \mu = \frac{a+b}{2} \] For example, with the rainfall in Flagstaff where the bounds are 0.5 and 3.0 inches, the calculation looks like this: \[ \mu = \frac{0.5 + 3.0}{2} = 1.75 \; \text{inches} \] This measure of central tendency tells us that the average expected rainfall is 1.75 inches, providing a useful summary of the distribution.

Standard Deviation

In statistics, standard deviation is used to quantify the amount of variation or dispersion in a set of data values. For a uniform distribution, the data is spread out evenly, making the calculation straightforward.

To find the standard deviation, use the formula: \[ \sigma = \frac{b-a}{\sqrt{12}} \] This helps measure how much the values in the distribution deviate from the mean on average.

Applying the values for our rainfall example (a = 0.5 and b = 3.0): \[ \sigma = \frac{3.0 - 0.5}{\sqrt{12}} \approx 0.7217 \; \text{inches} \] This result tells us that, on average, the amount of rainfall varies by about 0.72 inches from the mean. Lower standard deviation indicates less variability, and vice versa.

Probability

Probability in a uniform distribution tells us the chance of a random event happening within certain bounds. The uniform distribution implies all outcomes in the range are equally likely.

For instance, the probability of receiving less than 1 inch of rain is computed using the formula: \[ P(X < x) = \frac{x-a}{b-a} \] By plugging in the values a = 0.5 and x = 1.0, we get: \[ P(X < 1.0) = \frac{1.0 - 0.5}{3.0 - 0.5} = 0.2 \] Meaning there is a 20% chance the rainfall is less than 1 inch.

The chance of having exactly 1 inch is 0 since a specific point has no 'thickness' in continuous distributions, thus making exact probabilities zero.

Continuous Distribution

Unlike discrete distributions that deal with distinct separate values, continuous distributions have infinitely many possible values within a range. This continuous nature means we can't assign a positive probability to single points, only ranges.

A uniform continuous distribution, like our rainfall example, means every single value has an equal chance of occurring. So, calculating probabilities over intervals is what matters here. Consider the probability of more than 1.5 inches of rain, using: \[ P(X > x) = \frac{b-x}{b-a} \] Substituting b = 3.0 and x = 1.5 gives us: \[ P(X > 1.5) = \frac{3.0 - 1.5}{3.0 - 0.5} = 0.6 \]

This indicates a 60% likelihood of receiving more than 1.5 inches, a practical application of understanding continuous distributions.