5.3 Practical example

Now, you know how to calculate Z critical value. Let us do a z-test together.

Consider the following question:

A digital marketing agency has recently implemented a new advertising campaign aimed at increasing purchases on their e-commerce platform.

The marketing team has gathered data from a sample of 80 customers to assess the campaign’s effectiveness. The marketing executive asserts that the campaign has been successful, claiming that the average number of purchases per day has increased compared to the previous average. They want to test this claim with a 95% confidence level. Download the Excel file to review the data. Excel file: Digital marketing

Your task is to conduct a z-test to evaluate the marketing executive’s claim using the provided digital marketing dataset.

Let us solve this problem step by step using Excel:

Step 1: Find the Population Mean

First, we need to calculate the average of monthly purchases. In Excel:

• Go to the ‘Population’ sheet.
• Use the formula: =AVERAGE(A2:A1001) where A2:A1001 contains the population data.

The result shows the average number of purchases was 50.20. This is our population mean.

Step 2: Formulate Hypotheses

Based on the marketing executive’s claim, we can formulate our hypotheses:

• H₀: The digital marketing campaign will produce equal or less than average monthly purchase results (µ ≤ 50.20).

• H₁: The digital marketing campaign will produce greater than average monthly purchase results (µ > 50.20).

Note that we specify the claim we want to prove as an alternative hypothesis.

Step 3: Calculate the Z-statistic

We use the formula:

Where:

• = sample mean
• μ = population mean under the null hypothesis
• σ = population standard deviation
• n = sample size

To find these values:

1. Calculate the sample mean:

• Go to the ‘Sample’ sheet.
• Use the formula: =AVERAGE(A2:A81)

This gives us = 53.03

2. We already know µ = 50.20 from Step 1.

3. Calculate the population standard deviation:

• Go back to the ‘Population’ sheet.
• Use the formula: =STDEV.P(A2:A1001)

This gives us σ = 9.81

Important note on standard deviation functions:

Excel provides two functions for calculating standard deviation: STDEV.P() and STDEV.S().

• Use STDEV.P() when your data represents the entire population.
• Use STDEV.S() when your data is just a sample of the population.

In this case, we use STDEV.P() because we have data for the entire population of purchases.

4. We know n = 80 from the question and from our data set.

Now, let us calculate the z-statistic:

Step 4: Determine the Z Critical Value

For an upper tailed test, α = 5%, z critical value corresponds to the boundary = 95% of the z-distribution:

• In Excel, use the formula: =NORM.S.INV(0.95)

This gives us a z-critical value of 1.65.

Step 5: Make a Decision

Compare the calculated z-statistic (2.58) to the critical value (1.65). Since 2.58 > 1.65, we reject the null hypothesis. Figure 7 illustrates these results. You can see that z-statistic is outside of the fail to reject region and within the rejection region (orange area).

Step 6: Interpret the Results

There is sufficient statistical evidence to support the marketing executive’s claim that the new advertising campaign has increased the average number of monthly purchases. The sample data suggests that the true population mean of monthly purchases after the campaign is significantly higher than the population average of 50.20, at a 95% confidence level.