Listed below are paired data consisting of amounts spent on advertising (in millions of dollars) and the profits (in millions of dollars). Determine if there is a significant positive linear correlation between advertising cost and profit . Use a significance level of 0.01 and round all values to 4 decimal places.

MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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The content presents a statistical hypothesis testing exercise concerning the correlation between advertising expenses and profits. It includes a table, hypotheses, calculations, and conclusions, outlined as follows:

### Data Table
- The table lists paired values of advertising expenses and profits.

| Expense | Profit |
|---------|--------|
| 5       | 26     |
| 6       | 30     |
| 7       | 32     |
| 8       | 26     |
| 9       | 33     |
| 10      | 29     |
| 11      | 24     |

### Hypotheses
- Null Hypothesis (Ho): ρ = 0
- Alternative Hypothesis (Ha): ρ > 0

### Calculations
- **Find the Linear Correlation Coefficient** 
  - Formula: \( r = \, \) (blank space for input)

- **Find the p-value**
  - Formula: \( \text{p-value} = \, \) (blank space for input)

### Decision Making
- **The p-value is** 
  - Greater than α
  - Less than (or equal to) α

- **The p-value leads to a decision to** 
  - Do Not Reject Ho
  - Reject Ho
  - Accept Ho

### Conclusion
- **The conclusion is** 
  - There is insufficient evidence to make a conclusion about the linear correlation between advertising expense and profit.
  - There is a significant linear correlation between advertising expense and profit.
  - There is a significant negative linear correlation between advertising expense and profit.
  - There is a significant positive linear correlation between advertising expense and profit.
Transcribed Image Text:The content presents a statistical hypothesis testing exercise concerning the correlation between advertising expenses and profits. It includes a table, hypotheses, calculations, and conclusions, outlined as follows: ### Data Table - The table lists paired values of advertising expenses and profits. | Expense | Profit | |---------|--------| | 5 | 26 | | 6 | 30 | | 7 | 32 | | 8 | 26 | | 9 | 33 | | 10 | 29 | | 11 | 24 | ### Hypotheses - Null Hypothesis (Ho): ρ = 0 - Alternative Hypothesis (Ha): ρ > 0 ### Calculations - **Find the Linear Correlation Coefficient** - Formula: \( r = \, \) (blank space for input) - **Find the p-value** - Formula: \( \text{p-value} = \, \) (blank space for input) ### Decision Making - **The p-value is** - Greater than α - Less than (or equal to) α - **The p-value leads to a decision to** - Do Not Reject Ho - Reject Ho - Accept Ho ### Conclusion - **The conclusion is** - There is insufficient evidence to make a conclusion about the linear correlation between advertising expense and profit. - There is a significant linear correlation between advertising expense and profit. - There is a significant negative linear correlation between advertising expense and profit. - There is a significant positive linear correlation between advertising expense and profit.
**Educational Content: Analyzing Correlation Between Advertising Costs and Profits**

**Introduction:**
The table below presents paired data that includes the amounts spent on advertising (in millions of dollars) and the corresponding profits (also in millions of dollars). The goal is to determine if a significant positive linear correlation exists between advertising cost and profit using a significance level of 0.01. All calculations should be rounded to four decimal places.

**Data Table:**

| Advertising Cost (in millions) | Profit (in millions) |
|-------------------------------|----------------------|
| 3                             | 13                   |
| 4                             | 24                   |
| 5                             | 16                   |
| 6                             | 23                   |
| 7                             | 25                   |
| 8                             | 26                   |
| 9                             | 33                   |
| 10                            | 29                   |
| 11                            | 24                   |

**Hypotheses:**
- Null Hypothesis (Ho): ρ = 0 (There is no linear correlation between advertising cost and profit)
- Alternative Hypothesis (Ha): ρ > 0 (There is a positive linear correlation between advertising cost and profit)

**Analysis Steps:**

1. **Find the Linear Correlation Coefficient (r):**
   - Calculate the value of "r" that measures the strength and direction of the linear relationship between the two variables.

2. **Find the p-value:**
   - Determine the probability (p-value) associated with the observed value of "r" to assess the strength of the evidence against the null hypothesis.

**Decision Rule:**

- Compare the p-value to the significance level (α = 0.01).
- If the p-value is less than or equal to α, reject the null hypothesis in favor of the alternative hypothesis.
- If the p-value is greater than α, do not reject the null hypothesis.

**Conclusion:**

Based on your calculations, interpret the results to determine if there is significant evidence of a positive linear correlation between advertising costs and profits.
Transcribed Image Text:**Educational Content: Analyzing Correlation Between Advertising Costs and Profits** **Introduction:** The table below presents paired data that includes the amounts spent on advertising (in millions of dollars) and the corresponding profits (also in millions of dollars). The goal is to determine if a significant positive linear correlation exists between advertising cost and profit using a significance level of 0.01. All calculations should be rounded to four decimal places. **Data Table:** | Advertising Cost (in millions) | Profit (in millions) | |-------------------------------|----------------------| | 3 | 13 | | 4 | 24 | | 5 | 16 | | 6 | 23 | | 7 | 25 | | 8 | 26 | | 9 | 33 | | 10 | 29 | | 11 | 24 | **Hypotheses:** - Null Hypothesis (Ho): ρ = 0 (There is no linear correlation between advertising cost and profit) - Alternative Hypothesis (Ha): ρ > 0 (There is a positive linear correlation between advertising cost and profit) **Analysis Steps:** 1. **Find the Linear Correlation Coefficient (r):** - Calculate the value of "r" that measures the strength and direction of the linear relationship between the two variables. 2. **Find the p-value:** - Determine the probability (p-value) associated with the observed value of "r" to assess the strength of the evidence against the null hypothesis. **Decision Rule:** - Compare the p-value to the significance level (α = 0.01). - If the p-value is less than or equal to α, reject the null hypothesis in favor of the alternative hypothesis. - If the p-value is greater than α, do not reject the null hypothesis. **Conclusion:** Based on your calculations, interpret the results to determine if there is significant evidence of a positive linear correlation between advertising costs and profits.
Expert Solution
Step 1

Correlation:

The correlation value is obtained using EXCEL. The software procedure is given below:

  • Enter the data.
  • Select Data > Data Analysis >Correlation> OK.
  • Enter Input Range as A1:B11.
  • Mark Labels in First Row.
  • Click OK.

The output using EXCEL is as follows:

Statistics homework question answer, step 1, image 1

From the output, the correlation coefficient is 0.7204.

Thus, the linear correlation coefficient is 0.7204.

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