Exploring Standard Normal Distribution Activity Cosmic Coffee Shop Pricing

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Introduction

In this activity, we will delve into the fascinating world of the standard normal distribution and apply it to a real-world scenario: the pricing at the Cosmic Coffee shop. As discussed in the introduction, the Cosmic Coffee shop offers a variety of beverages, and their prices follow a normal distribution. Understanding this distribution allows us to answer various questions about the likelihood of certain price ranges, the percentage of drinks falling within specific price points, and other insightful statistical inquiries. The mean price of a drink at Cosmic Coffee is $4.25, with a standard deviation of $0.67. This information is crucial as we navigate through the following questions and utilize the standard normal table (also known as the z-table) to find probabilities and percentiles.

The standard normal distribution, a cornerstone of statistics, is a normal distribution with a mean of 0 and a standard deviation of 1. It is a powerful tool because any normal distribution can be transformed into a standard normal distribution, allowing us to use a single table (the z-table) to find probabilities for any normally distributed variable. The z-table provides the area under the standard normal curve to the left of a given z-score. The z-score represents the number of standard deviations a particular value is away from the mean. By calculating z-scores and consulting the z-table, we can determine the probability of observing a value within a certain range or above/below a specific value. This has wide-ranging applications in fields like finance, engineering, and, in our case, business and pricing analysis.

The following questions will challenge you to apply your understanding of the standard normal distribution and the z-table to analyze the prices at Cosmic Coffee. You will need to calculate z-scores, use the z-table to find probabilities, and interpret these probabilities in the context of the coffee shop's pricing strategy. Remember that the mean price of $4.25 and the standard deviation of $0.67 are your key pieces of information. As you work through the problems, focus on understanding what each question is asking, how to translate the question into a statistical problem, and how to use the z-table effectively. This activity is designed to strengthen your understanding of statistical concepts and demonstrate their practical application in a business setting. Embrace the challenge, and let's explore the pricing landscape of Cosmic Coffee through the lens of the standard normal distribution.

Question 1: Part A

Part A

Discussion Category: Mathematics

This part of the activity likely involves a specific question or problem related to the Cosmic Coffee shop pricing, requiring mathematical calculations and statistical reasoning. Given the context of the activity, the question will undoubtedly involve the standard normal distribution, the mean price of $4.25, the standard deviation of $0.67, and the use of the standard normal table (z-table). The mathematical focus suggests that you will need to perform calculations such as finding z-scores, looking up probabilities in the z-table, and potentially calculating percentiles or ranges of prices. It's also possible that the question might involve inverse calculations, where you are given a probability and need to find the corresponding price or z-score.

To effectively tackle this part, it's essential to carefully read and understand the specific question being asked. Identify the key information provided, such as a target price, a price range, or a probability target. Then, determine the appropriate steps to solve the problem. This might involve calculating the z-score for a given price using the formula: z = (X - μ) / σ, where X is the price, μ is the mean price, and σ is the standard deviation. Once you have the z-score, you can consult the z-table to find the corresponding probability. Remember that the z-table typically provides the area to the left of the z-score. If the question asks for the area to the right, you will need to subtract the z-table value from 1. If the question involves a range of prices, you will need to calculate z-scores for both endpoints of the range and find the difference between their corresponding probabilities.

The mathematical category also implies that accuracy is crucial. Ensure that you perform calculations correctly and pay attention to decimal places when looking up values in the z-table. Double-check your work to avoid errors that could lead to incorrect answers. Furthermore, it's important to interpret your results in the context of the problem. Don't just provide a numerical answer; explain what the answer means in terms of the Cosmic Coffee shop's pricing. For example, if you find the probability of a drink costing less than a certain amount, state your answer in a way that is understandable and relevant to the situation. This demonstrates a thorough understanding of the concepts and their application. Ultimately, approaching this part with a clear understanding of the mathematical principles underlying the standard normal distribution and careful attention to detail will be key to success.

This part could ask questions like:

  • What is the probability that a randomly selected drink at Cosmic Coffee costs less than $3.50?
  • What is the probability that a drink costs more than $5.00?
  • What percentage of drinks cost between $4.00 and $4.50?
  • What is the price that separates the top 10% of most expensive drinks?

To answer these questions, you would follow these general steps:

  1. Understand the Question: Clearly identify what the question is asking.
  2. Calculate the Z-score: Use the formula z = (X - μ) / σ, where X is the value in question, μ is the mean ($4.25), and σ is the standard deviation ($0.67).
  3. Use the Z-table: Look up the calculated z-score in the standard normal table to find the corresponding probability.
  4. Interpret the Result: Translate the probability from the z-table into an answer that makes sense in the context of the problem.

By practicing these types of questions, you can become more confident in your ability to apply the standard normal distribution to real-world scenarios and make informed decisions based on statistical analysis.