Question

The daily sales total (excepting Saturday) at a small restaurant has a probability distribution that is approximately normal, with a mean \(\mu\) equal to \(\$ 1230\) per day and a standard deviation \(\sigma\) equal to \(\$ 120\). a. What is the probability that the sales will exceed \(\$ 1400\) for a given day? b. The restaurant must have at least \(\$ 1000\) in sales per day to break even. What is the probability that on a given day the restaurant will not break even?

Short answer

Answer: The probability for the restaurant to exceed sales of $1400 on a given day is approximately 7.78%, while the probability for the restaurant not to break even (sales less than $1000) on a given day is approximately 2.74%.

Step-by-step solution

Identify the given information

We are given the mean (\(\mu\)) and standard deviation (\(\sigma\)) of the daily sales: Mean, \(\mu = \$1230\) Standard Deviation, \(\sigma = \$120\)

Find the z-score for each question

To find the probabilities, we first need to find the z-scores for each sales amount. a. Sales exceeding $1400: z = \frac{X - \mu}{\sigma} = \frac{1400 - 1230}{120} = \frac{170}{120} = 1.42$ b. Sales not breaking even (less than $1000): z = \frac{X - \mu}{\sigma} = \frac{1000 - 1230}{120} = \frac{-230}{120} = -1.92$

Find the corresponding probabilities

Now, we'll find the probabilities using a z-table or standard normal distribution calculator. a. The z-score for sales exceeding \(1400 is 1.42. We find the area to the right of this z-score (since we want the probability of sales exceeding \)1400): P(Z > 1.42) = 1 - P(Z < 1.42) = 1 - 0.9222 = 0.0778$ b. The z-score for sales not breaking even (less than \(1000) is -1.92. We find the area to the left of this z-score (since we want the probability of sales being less than \)1000): P(Z < -1.92) = 0.0274$

Interpret the results

Here are the answers to the two questions: a. The probability that the sales will exceed $1400 for a given day is approximately 0.0778 or 7.78%. b. The probability that on a given day the restaurant will not break even (sales less than $1000) is approximately 0.0274 or 2.74%.

Key concepts

Z-score Calculation

Understanding the z-score is crucial when working with normal distributions in probability and statistics. Essentially, a z-score measures how many standard deviations an element is from the mean. To calculate a z-score, we use the formula:
\[\begin{equation} z = \frac{X - \mu}{\sigma} \end{equation}\]
In this formula, \(X\) represents the value of interest, \(\mu\) is the mean of the dataset, and \(\sigma\) is the standard deviation. The calculated z-score tells us if the value \(X\) is above or below the mean, and by how much. For example, a z-score of 1.42 indicates that a value is 1.42 standard deviations above the mean. Similarly, a negative z-score, like -1.92, implies that the value is 1.92 standard deviations below the mean.
Z-scores help compare different data points from different normal distributions or assess probabilities in the standard normal distribution.

Importance of Z-score in Probability

Z-scores convert individual data points to a common scale, enabling us to make probabilistic predictions. They assist in determining how unusual or commonplace a particular result is within the context of a given distribution. For students, it's important to practice converting raw scores to z-scores to become proficient in evaluating probabilities in various statistical scenarios.

Standard Normal Distribution

The standard normal distribution, an essential concept in statistics, is a specific case of the normal, or Gaussian, distribution with a mean of zero and a standard deviation of one. This distribution is represented by the bell curve, where approximately 68% of all data falls within one standard deviation from the mean, increasing to 95% within two standard deviations, and about 99.7% within three.
What makes the standard normal distribution so powerful is its use in the z-score calculations. Once you have a z-score, you can refer to the standard normal distribution to find out the probability of a score occurring within a given dataset.

• The area to the left of the z-score provides the probability of obtaining a score less than our value.
• The area to the right indicates the probability of getting a score greater than the one in question.

In practice, standard normal distribution tables or software can be used to find these probabilities without having to perform complex integrations every time. Knowing how to read these tables or use these tools is critical for students as they allow us to quickly assess probabilities and make informed decisions based on statistical data.

Probability and Statistics

Probability and statistics are branches of mathematics that deal with data analysis, interpretation, collection, and presentation. At the heart of statistics is the concept of understanding and quantifying uncertainty through probability.

Differentiating Between Probability and Statistics

• Probability is the study of random events and is used to predict how likely they are to occur. It is the groundwork on which statistical theory is built.
• Statistics takes concepts from probability and applies them to data in order to describe and infer patterns and relationships.

Applications in Real Life

Probability and statistics are applied in diverse fields, from predicting weather patterns to decision making in business and economics. They are also used to test hypotheses in research and to inform public policy.
For students tackling problems like the daily sales of a restaurant, understanding probability and statistics is crucial. They will encounter concepts such as the normal distribution, z-scores, and hypothesis testing. The ability to integrate and apply these tools can help them not only solve homework problems but also interpret real-world data effectively.