When performing a x²-test for independence, what is the test statistic?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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Please answer the question from the image
Transcribed Image Text:### Survey on Preferred Shipping Carriers for Holiday Gifts
A local news survey asked 500 randomly sampled people about which shipping carrier they prefer to use for shipping holiday gifts. The table below shows the distribution of responses by age group. Perform a chi-square hypothesis test for independence and answer the following.
#### Distribution of Preferred Shipping Carriers by Age Group
| CARRIER | 18-34 | 35-54 | 55+ |
|-----------------|-------|-------|-----|
| **USPS** | 72 | 97 | 76 |
| **UPS** | 52 | 76 | 34 |
| **FedEx** | 31 | 24 | 9 |
| **Something Else** | 7 | 6 | 3 |
| **Not Sure** | 3 | 6 | 4 |
The age groups are divided into:
- **18-34 years old**
- **35-54 years old**
- **55+ years old**
Each cell in the table represents the number of respondents from each age group that chose the corresponding shipping carrier. The carriers considered in the survey include USPS, UPS, FedEx, Something Else, and Not Sure. The goal is to determine whether there is a significant association between age group and preferred shipping carrier.
#### Instructions for Statistical Analysis
Perform a chi-square test for independence using the given data to evaluate whether the preference for shipping carriers varies significantly across different age groups.
![### Understanding the Chi-Square Test for Independence
---
**Question:** When performing a χ²-test for independence, what is the test statistic?
---
### Explanation:
The chi-square (χ²) test for independence is used to determine whether there is a significant relationship between two categorical variables. The test statistic for the chi-square test for independence is denoted by χ².
The test statistic is calculated using the formula:
\[
\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}
\]
where:
- \(O_i\) represents the observed frequency (the data collected).
- \(E_i\) represents the expected frequency (the frequency we would expect if there is no relationship between the variables).
- The sum is calculated over all possible categories.
### Interpretation:
- **High χ² value:** Suggests a higher discrepancy between observed and expected frequencies, indicating that the variables are likely dependent.
- **Low χ² value:** Suggests a lower discrepancy between observed and expected frequencies, indicating that the variables are likely independent.
### Conclusion:
In the context of hypothesis testing, if the calculated χ² statistic is greater than the critical value from the Chi-Square distribution table (based on the degrees of freedom and the chosen significance level), we reject the null hypothesis of independence. Otherwise, we do not reject the null hypothesis.
Understanding the test statistic and how to interpret it is crucial for correctly using the chi-square test for independence in statistical analyses.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf70a0c0-4efb-4246-bce2-070135894775%2F77ab74b0-0508-4dca-b843-9edbbc5c907a%2Fjznrsrq_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding the Chi-Square Test for Independence
---
**Question:** When performing a χ²-test for independence, what is the test statistic?
---
### Explanation:
The chi-square (χ²) test for independence is used to determine whether there is a significant relationship between two categorical variables. The test statistic for the chi-square test for independence is denoted by χ².
The test statistic is calculated using the formula:
\[
\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}
\]
where:
- \(O_i\) represents the observed frequency (the data collected).
- \(E_i\) represents the expected frequency (the frequency we would expect if there is no relationship between the variables).
- The sum is calculated over all possible categories.
### Interpretation:
- **High χ² value:** Suggests a higher discrepancy between observed and expected frequencies, indicating that the variables are likely dependent.
- **Low χ² value:** Suggests a lower discrepancy between observed and expected frequencies, indicating that the variables are likely independent.
### Conclusion:
In the context of hypothesis testing, if the calculated χ² statistic is greater than the critical value from the Chi-Square distribution table (based on the degrees of freedom and the chosen significance level), we reject the null hypothesis of independence. Otherwise, we do not reject the null hypothesis.
Understanding the test statistic and how to interpret it is crucial for correctly using the chi-square test for independence in statistical analyses.
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