Confidence Intervals
Means vs Proportions
Learning Objectives
In this section, we will introduce the concept of estimating population proportions and understand how these different from population means.
Two Population Parameters to Estimate
ln this chapter, there are two population parameters that we will be investigating using sampling. They are:
- The true mean (or average) of a population, also symbolically known as [latex]\mu[/latex], and
- The true proportion of a population, also symbolically represented by [latex]p[/latex].
In the first five sections, we estimated the true mean of a population when [latex]\sigma[/latex] was known and unknown. The remaining sections in this chapter refer to estimating the true proportion of a population ([latex]p[/latex]).
The Difference Between a Population Mean and Proportion
The difference between a population mean and population proportion is quite simple. We are calculating a true mean, µ, when dealing with a measured variable, i.e., something that can be physically measured to a desired degree of precision. Variables of this type are called quantitative variables.
We are trying to understand the true percentage, [latex]p[/latex], when trying to measure a fraction of population or true proportion of a population. Variables or columns in the data where we record which trait, or category, a person or item belongs to, are called categorical variables.
Mean or Proportion? (Exercise #1)
Select which type each measurement is from the list below. Is it an example of quantitative (mean) data or categorical (proportion) data?
Click here to review the solutions to the above exercise
| Quantitative Variable (mean or μ) | Categorical Variable (proportion or [latex]p[/latex]) |
|---|---|
| Amount of rainfall in Vancouver (in mms) | Percent of rainy days per annum in Vancouver |
| Number of defective items in inventory | Percent of defective parts in inventory |
| Number of late deliveries per month | Percent of late deliveries per month |
| Number of absent employees per day | Percent of absent employees per day |
The table above provides the obvious distinction between the two forms of problems you will encounter: estimating a true mean of a population versus estimating a true proportion of population.
Mean or Proportion? (Exercise #2)
In the exercise below, move the metric to the correct side – categorical or quantitative. All of the statistics reported in the exercise above are reported by Hostinger Tutorials in the following article.
Click here to review the solutions to the above exercise
| Categorical Variable (proportion or [latex]p[/latex]) | Quantitative Variable (mean or μ) |
|---|---|
| More than 50% of company’s marketing budgets go to paid media and new technologies | Global Digital Advertising Market is Expected to Reach $740 Billion in 2024 |
| Around 84% of B2B marketers create brand awareness through content marketing | Uses spend 6 hours and 48 minutes, on average, watching online videos each week |
| On average, the first result on Google generates 39.8% of all clicks | The global market size of television advertising was worth $235 billion in 2023 |
| With featured snippets, the click-through rate increases to 42.9% | In 2023, the market value of digital marketing reach $366 billion |
| Nearly half of millenials believe influencer recommendations more than regular ads when buying a product | On average, companies invest $34.50 per internet user to create and promote video ads |
All of the statistics reported in the exercise above are reported by Hostinger Tutorials in the following article.
