Problem 5 - Basic Statistics for Business & Economics

Question

The information below is the number of daily emergency service calls made by the volunteer ambulance service of Walterboro, South Carolina, for the last 50 days. To explain, there were 22 days on which there were two emergency calls, and 9 days on which there were three emergency calls. $$ \begin{array}{|cc|} \hline \text { Number of Calls } & \text { Frequency } \\ \hline 0 & 8 \\ 1 & 10 \\ 2 & 22 \\ 3 & 9 \\ 4 & 1 \\ \hline \text { Total } & 50 \\ \hline \end{array} $$ a. Convert this information on the number of calls to a probability distribution. b. Is this an example of a discrete or continuous probability distribution? c. What is the mean number of emergency calls per day? d. What is the standard deviation of the number of calls made daily?

Step-by-Step Solution

Answer

Probabilities: 0.16, 0.20, 0.44, 0.18, 0.02; Discrete distribution; Mean: 1.70; Std. Dev: 1.00.

Step 1: Identify Frequencies and Total

The provided data in the table shows the number of emergency calls and their corresponding frequencies over 50 days. We will use this data to convert it into a probability distribution.

Step 2: Create Probability Distribution

To convert frequencies to probabilities, divide each frequency by the total number of days (50). - Probability of 0 calls: \( P(0) = \frac{8}{50} = 0.16 \)- Probability of 1 call: \( P(1) = \frac{10}{50} = 0.20 \)- Probability of 2 calls: \( P(2) = \frac{22}{50} = 0.44 \)- Probability of 3 calls: \( P(3) = \frac{9}{50} = 0.18 \)- Probability of 4 calls: \( P(4) = \frac{1}{50} = 0.02 \) This forms our probability distribution for the number of emergency calls.

Step 3: Determine Type of Probability Distribution

Since the information provided lists a finite number of distinct outcomes (0, 1, 2, 3, 4 calls) with respective probabilities, this is a discrete probability distribution.

Step 4: Calculate the Mean (Expected Value)

The mean of a probability distribution is calculated as \( \mu = \sum (x_i \times P(x_i)) \), where \( x_i \) is the number of calls and \( P(x_i) \) is their probability. \[ \begin{align*} \mu &= 0 \times 0.16 + 1 \times 0.20 + 2 \times 0.44 + 3 \times 0.18 + 4 \times 0.02 \ &= 0 + 0.20 + 0.88 + 0.54 + 0.08 \ &= 1.70\end{align*} \]The mean number of calls per day is 1.70.

Step 5: Calculate the Variance

Variance is calculated as \( \sigma^2 = \sum ((x_i - \mu)^2 \times P(x_i)) \).\[ \begin{align*}\sigma^2 &= (0 - 1.7)^2 \times 0.16 + (1 - 1.7)^2 \times 0.20 + (2 - 1.7)^2 \times 0.44 \ &\quad + (3 - 1.7)^2 \times 0.18 + (4 - 1.7)^2 \times 0.02 \ &= 2.89 \times 0.16 + 0.49 \times 0.20 + 0.09 \times 0.44 \ &\quad + 1.69 \times 0.18 + 5.29 \times 0.02 \ &= 0.4624 + 0.098 + 0.0396 + 0.3042 + 0.1058 \ &= 1.01\end{align*} \]The variance of the number of calls is 1.01.

Step 6: Calculate the Standard Deviation

Standard deviation is the square root of the variance: \( \sigma = \sqrt{1.01} \approx 1.00 \).Thus, the standard deviation of the number of daily emergency calls is approximately 1.00.

Key Concepts

Discrete Probability

Discrete probability distribution is a way to describe the likelihood of various outcomes when dealing with discrete random variables. A discrete random variable is one where the outcomes can be counted distinctly, like the number of calls made daily by an emergency service, which can only be 0, 1, 2, 3, 4, etc.

In a discrete probability distribution, each outcome is assigned a probability, showing how likely each event is to occur. These probabilities are calculated by dividing the frequency of each outcome by the total number of observations. For example, if the probability of receiving 0 calls in a day is 0.16, this means there's a 16% chance that no calls will happen on any given day. The sum of all probabilities in a discrete distribution must always equal 1.0, as they cover all possible outcomes.

Understanding discrete probability distributions allows us to model real-world situations where outcomes are limited to specific, countable possibilities. This can be incredibly useful in planning and predicting different scenarios and in making informed decisions based on statistical data.

Expected Value

The expected value, also known as the mean, helps us understand the 'average' outcome in a probability distribution. Although it might not correspond to an actual outcome, it gives us a central value or a point of balance for the data.

For a discrete probability distribution, the expected value is calculated by multiplying each outcome by its respective probability and then summing up all these products. Mathematically, this can be represented as:

\( \mu = \sum (x_i \times P(x_i)) \)

where \(x_i\) is an individual value, and \(P(x_i)\) is its probability. Using the data from the exercise, the expected number of calls per day was found to be 1.70.

This means that, on average, the ambulance service can expect about 1.70 calls per day over a long period. The expected value provides valuable insight into what is typically seen in the data, helping organizations, like the ambulance service, allocate resources efficiently based on predicted trends.

Variance and Standard Deviation

Variance and standard deviation are statistical measures used to describe the variability or spread of data in a probability distribution. They provide insight into the consistency or predictability of outcomes.

**Variance** measures how far each outcome deviates from the expected value, averaged over all possible outcomes. It's calculated by taking the average of the squared differences between each outcome and the expected value:

\( \sigma^2 = \sum ((x_i - \mu)^2 \times P(x_i)) \)

In the ambulance call data example, the variance was calculated to be 1.01. This number alone might not be very intuitive.

**Standard Deviation** is the square root of the variance, offering a more understandable metric since it has the same unit as the data itself. In our example, the standard deviation was approximately 1.00.

Both measures are crucial as they indicate the level of variability in daily call volumes. A higher standard deviation would mean more unpredictability and inconsistency in call numbers, while a lower standard deviation suggests more consistent results close to the expected value. This knowledge helps organizations prepare for fluctuations and manage resources effectively.