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Data analysis: hypothesis testing

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# 5.3 Using the z-score

So far, you have identified the range of z-scores that indicate the reject regions for the hypothesis tests.

For example, records indicate that customers need to be exposed to TV advertising commercials for an average of 10 seconds before being influenced by the advertisement, with a standard deviation of 1.6 seconds. A marketing manager suggests that it will take longer to influence customer behaviour. The z-score table can be used to identify the reject region (for the null hypothesis) in the right tail where the z-score is greater than 1.28. This will test the marketing manager’s claim at a 90% confidence level (α = 0.10) with a sample of 100 customers.

Figure 21 Advertisement influence

However, what does this mean in practice? Marketing managers are not particularly concerned with the z-score indicating the hypothesis’ cut-off region. Rather, they want to know how long television advertisements should be on for on average in order to effectively influence customer behaviour with 90% confidence.

In order to answer this question, you need the formula for calculating the z-score.

= sample mean

= population mean

= population standard deviation

n = sample size

In the example used above, you know that µ equals 10 (TV advertising commercials are on for an average of 10 seconds before customers are influenced by the advertisement) and σ equals 1.6. You also know that the rejection region of the z-score is > 1.28, and the sample size equals 100 (a sample of 100 customers).

These values can be input into the z-score formula to solve the value of x.

Step 1:

Step 2:

Step 3:

Step 4:

In the end, you can see that the sample mean () needs to be 10.2. In the decision rule, you can state that in terms of obtaining a z-score of 1.28, you will reject the null hypothesis for any sample mean above 10.2.

In other words, if a marketing manager decides to conduct market research by surveying 100 customers about their responses to TV advertising commercials, they will reject the general belief – customers need to be exposed to TV advertising commercials on average 10 seconds before being influenced by the advertisement – when they find that the average time taken for TV advertising commercials to influence customers in this sample is greater than 10.2 seconds.

In the following activity, you will be tested on your understanding of how to calculate the mean of a sample, which is used to reject the null hypothesis.

## Activity 7 Determine the range of mean used to reject null hypothesis

Timing: Allow approximately 30 minutes to complete this activity

Calculating the mean is a key statistical analysis technique used to summarise data and draw conclusions. In hypothesis testing, the mean is crucial in determining whether to reject the null hypothesis, which assumes no significant difference between two variables. To reject this hypothesis, a statistical test is used to compare the mean of a sample to a known or expected value.

This activity aims to provide a comprehensive understanding of mean calculation and how to use it to reject the null hypothesis. By the end of the activity, you’ll be able to perform calculations, interpret statistical results and make informed decisions based on the mean.

According to records, customers are persuaded to purchase the products by a price discount of 50% in marketing promotions with a standard deviation of 5.3%. The market research team believes that the company should offer more price discounts (more than 50%) to motivate customers’ purchase intentions. To test this claim, the team will survey 1000 customers. At 99% confidence (significance) levels, can you state the decision rule concerning the value of mean to accept the claim made by the marketing team?

Use the free response box below to show your calculations.

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### Discussion

Step 1: State the hypotheses.

H0: Customers will be persuaded to purchase the products by a price discount of less than or equal to 50% in a marketing promotion (µ ≤ 50% price discount).

Ha: Customers will be persuaded to purchase the products with a price discount of more than 50% in a marketing promotion (µ > 50% price discount).

Step 2: Identify the z-score and rejection region.

Using the z-score table, you can identify the z-score for a one-tailed test with 99% significant levels. In this case, the z-score equals 2.33.

Step 3: Identify all the values.

Step 4: Solve the from the formula

If

Then

Step 5: Write the decision rule concerning the value of mean.

The market research team will reject the general belief – customers will be persuaded to purchase the products by less than or equal to 50% price discount in marketing promotion – when survey results (from 1000 customers) show that customers are persuaded to purchase the products when price discount promotion is greater than 50%.

In conclusion, understanding the range of z-scores and mean calculation in hypothesis testing is essential for drawing accurate statistical conclusions. By calculating the z-score and mean of a sample, you can determine the level of significance and whether the null hypothesis should be rejected or not. Furthermore, understanding how to interpret and use the z-score and mean enables you to make informed decisions based on the results of statistical analysis. By applying these concepts, you can draw meaningful conclusions and make informed decisions in a variety of fields, from healthcare to economics.