https://file.aiquickdraw.com/mathbot/file/17272919796724fifntou.pdf

The problem from the provided document involves calculating statistics based on customer ratings for Skyology Inc. and performing sampling. Here's a breakdown of the key elements:

### 1. **Population Information**:
- **Customers**: A and B
- **Ratings**: A gives a rating of 6, and B gives a rating of 7.
- **Population mean (μ)**: \( 6.50 \)
- **Population standard deviation (σ)**: \( 0.50 \)

### 2. **Sampling Process**:
- Sample of size 3, chosen with replacement from customers A and B.
- All possible samples are listed (8 samples total, since \( 2^3 = 8 \)).

The goal is to:
1. **Complete the table** by calculating the sample means for each sample.
2. **Find the mean (μₓ) and standard deviation (σₓ) of the sampling distribution of the sample means**.

### Step-by-step solution:

#### (a) Completing the table:

Here are the sample ratings, and their respective means:
- **Sample 1 (A, A, A)**: \( (6, 6, 6) \), Mean: \( 6.00 \)
- **Sample 2 (A, A, B)**: \( (6, 6, 7) \), Mean: \( 6.33 \)
- **Sample 3 (A, B, A)**: \( (6, 7, 6) \), Mean: \( 6.33 \)
- **Sample 4 (A, B, B)**: \( (6, 7, 7) \), Mean: \( 6.67 \)
- **Sample 5 (B, A, A)**: \( (7, 6, 6) \), Mean: \( 6.33 \)
- **Sample 6 (B, A, B)**: \( (7, 6, 7) \), Mean: \( 6.67 \)
- **Sample 7 (B, B, A)**: \( (7, 7, 6) \), Mean: \( 6.67 \)
- **Sample 8 (B, B, B)**: \( (7, 7, 7) \), Mean: \( 7.00 \)

#### (b) Calculating the Mean (μₓ) and Standard Deviation (σₓ) of the Sample Means:

- **Mean of the sample means (μₓ)**:  
  \[
  μₓ = \frac{6.00 + 6.33 + 6.33 + 6.67 + 6.33 + 6.67 + 6.67 + 7.00}{8} = 6.50
  \]
  So, \( μₓ = 6.50 \), which equals the population mean as expected in a sampling distribution.

- **Standard deviation of the sample means (σₓ)**:
  \[
  σₓ = \sqrt{\frac{(6.00 - 6.50)^2 + (6.33 - 6.50)^2 + \dots + (7.00 - 6.50)^2}{8}}
  \]
  The standard deviation was computed as \( 0.29 \).

#### (c) Graphing the Frequency Histograms:

- **Population distribution**:
  - Only two ratings (6 and 7), each with a frequency of 1.
  
- **Sampling distribution of the sample mean**:
  - The sample means: 6.00, 6.33, 6.67, and 7.00.
  - The frequency of each sample mean can be plotted.

Would you like to go into more detail on any part of this problem?

Here are five related questions you could explore:
1. How is the standard deviation of the sampling distribution related to the population standard deviation?
2. What are the implications of sampling with or without replacement?
3. How would the results change with a larger sample size?
4. Can this approach be extended to a larger population of customers?
5. What is the central limit theorem, and how does it apply here?

**Tip**: The standard deviation of the sampling distribution of the sample mean decreases as the sample size increases.

The problem involves calculating statistics based on customer ratings for Skyology Inc., including sampling, population mean, and standard deviation. Two customers, A and B, provided ratings, and a sample of size 3 was chosen with replacement. The goal is to calculate the sample means, the mean (μₓ), and the standard deviation (σₓ) of the sampling distribution.

Learn how to calculate the sample means, mean (μₓ), and standard deviation (σₓ) of the sampling distribution based on customer ratings for Skyology Inc. Includes a step-by-step solution with relevant formulas and a graph of the sampling distribution. Suitable for advanced high school or college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation