Question

Sample survey. A polling organization is checking its database to see if the two data sources it used sampled the same zip codes. The variable Datasource \(=1\) if the data source is MetroMedia, 2 if the data source is DataQwest, and 3 if it's RollingPoll. The organization finds that the correlation between five-digit zip code and Datasource is \(-0.0229\). It concludes that the correlation is low enough to state that there is no dependency between Zip Code and Source of Data. Comment.

Short answer

The correlation suggests no linear dependency, but correlation is not always the best metric for categorical data.

Step-by-step solution

Understand Correlation Coefficient

A correlation coefficient measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1. Values close to 0 suggest a weak linear relationship.

Analyze Given Correlation

The given correlation between zip code and data source is -0.0229. This value is very close to 0, indicating a very weak or no linear relationship between the two variables.

Interpret the Correlation for Dependency

Since the correlation is close to zero, it is reasonable to conclude that there is no linear dependency between zip code and data source on a numerical basis. However, correlation does not imply causation or dependency in all cases.

Consider the Nature of Data

Zip codes and data sources are categorical variables, and correlation might not be the best measure for dependency in such cases. Alternative tests, like Chi-square test for independence, may be more appropriate.

Key concepts

Sample Survey

A sample survey is a research method used to gather data from a select group of individuals out of a larger population. This technique allows researchers to make inferences about the larger group based on the observations from the sample. The key to a successful sample survey is ensuring that the sample is representative of the population. This helps in obtaining reliable and valid results. For example, if a polling organization wants to know which data sources are commonly used in certain zip codes, they might conduct a sample survey by selecting a portion of zip codes and checking which data source is associated with each. The results can provide insights, as long as the chosen sample reflects the diversity and characteristics of the whole population accurately. To conduct an effective sample survey:

• Define your target population clearly.
• Determine the sample size that will provide enough data points to be statistically significant.
• Choose an appropriate sampling method, such as random sampling, to avoid bias.
• Ensure the data collection process is standardized to reduce errors.

Using a sample survey is a cost-effective way to collect data when surveying the entire population isn't feasible. However, careful planning is essential to avoid biases and inaccuracies.

Categorical Variables

Categorical variables are types of data that can be divided into distinct categories or groups. They don't have a numerical value that implies order or quantity but rather describe characteristics or traits. For instance, the data source in the exercise is a categorical variable, with categories like MetroMedia, DataQwest, and RollingPoll represented by numbers 1, 2, and 3. When dealing with categorical variables, it's important to remember that:

• Each category represents a different group or type.
• Numbers assigned to different categories are arbitrary and don't have mathematical meanings.
• Analyzing categorical data often requires statistical tests suited for category-based comparisons, such as the Chi-square test.

Using the Chi-square test can help determine if there is a significant association between two categorical variables, challenging to discern through correlation analysis alone which measures linear relationships.

Chi-square Test

The Chi-square test is a statistical method used to determine if there is a significant association between two categorical variables. It helps in figuring out whether the observed differences in sample data could have occurred by chance, or if there is an actual relationship present between the variables.When applying the Chi-square test:

• Construct a contingency table displaying the frequency counts of categories in each variable.
• Calculate the expected frequency for each category assuming no association exists.
• Use the Chi-square formula: \[ \ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \] where \( O_i \) is the observed frequency and \( E_i \) is the expected frequency.
• Determine the p-value to infer statistical significance.

The Chi-square test is particularly powerful for hypothesis testing when you want to check for dependence between two categorical variables, making it an alternative or complement to correlation analysis in scenarios where variables do not have linear relationships.