7.1 Practical example

To illustrate the concept of proportions, let us revisit our digital marketing dataset containing 80 customers in a sample. In addition to the monthly purchases data, we also have a column labelled "big customer". This column identifies customers who have made over 55 purchases per month. In this sample, we find that 32 customers are identified as big customers.

Using this dataset, we can demonstrate how to calculate and interpret proportions:

• Identify the specific group: Our group of interest is “big customers”.

• Count the number of items in the group: We have 32 customers classified as “big customers”.

• Calculate the proportion: We divide the number of big customers by the total sample size.

• Proportion = = 0.4 or 40%

This proportion tells us that 40% of the customers in our sample are considered "big customers" based on our definition.

We can use this proportion for various analytical purposes:

1. Describing the customer base: We can state that two-fifths of our sampled customers are high-volume purchasers, making over 55 purchases per month.

1. Segmentation analysis: This proportion could inform marketing strategies, helping to tailor approaches for different customer segments. With 40% being big customers, we might consider developing specific retention strategies for this significant group.

1. Trend analysis: By calculating this proportion over time, we can track changes in the composition of our customer base. An increase in this proportion might indicate successful upselling strategies, while a decrease could signal a need for customer retention efforts.

1. Hypothesis testing: We might want to test whether the true population proportion of big customers differs from a hypothesised value. In this case, we would use a z-test for proportions.

Here, we are particularly interested in hypothesis testing. For example, a marketing executive reviews this data and formulates a claim that the percentage of big customers in the entire population is more than 30%. We want to test this claim using a hypothesis test.

Question: At a 95% confidence level, is there enough evidence to support the marketing executive’s claim that the population proportion of big customers is more than 30%?

We will use the following formula to test proportions:

p̂ = sample proportion

p₀ = population proportion under the null hypothesis

n = sample size

Step 1: Formulate the hypotheses

• H₀: p ≤ 0.30 (population proportion is 30% or less)
• H₁: p > 0.30 (population proportion is more than 30%)

Step 2: Determine the z-statistic

Given information:

• n = 80 customers
• p̂ = 0.4
• p₀ = 0.3

We use the z-test for proportions. The formula is:

Step 3: Determine the critical value

For an upper tailed test, at α = 5%, we use the Excel function: =NORM.S.INV(0.95)

• The z critical value is 1.645.

Step 4: Compare the z-statistic to the critical value and make a decision.

Our calculated z-statistic (1.952) is greater than the critical value (1.645).

Since the calculated z-statistic exceeds the critical value, we reject the null hypothesis.

Conclusion:

At a 95% confidence level, there is enough evidence to support the marketing executive’s claim that the population proportion of big customers is more than 30%. The sample data provides sufficient evidence to conclude that the true population proportion is significantly higher than 30%.