Two-Sample Hypothesis Testing for Means: Right-Tailed Test

1. Introduction to Two-Sample Hypothesis Tests for Means: Right-Tailed Tests

Two-sample hypothesis testing for means in right-tailed tests is a statistical method used to determine if the mean of one group is significantly greater than the mean of another group. In this type of test, the null hypothesis states that the means of the two groups are equal, while the alternative hypothesis suggests that the mean of the first group is greater than the mean of the second group. This approach is particularly useful when you want to test for improvements or increases in a specific metric.

For example, suppose you want to analyze the effectiveness of a new marketing strategy. You could compare the average sales figures from two different periods: one before the implementation of the new strategy and one after. The right-tailed test would help determine if the new strategy has significantly increased sales. The null hypothesis would be that the average sales before and after the strategy are the same, and the alternative hypothesis being that the average sales after the strategy are higher.

1a. Applications of Two-Sample Hypothesis Tests for Means: Right-Tailed Tests

Let us walk through a practical example of performing a two-sample hypothesis test for means for a right-tailed test.

EXAMPLE

BrightFuture Electronics, a company specializing in consumer electronics, recently launched a new marketing strategy aimed at boosting sales. To evaluate the effectiveness of this strategy, the marketing team decided to compare the average monthly sales figures from two different periods: three months before the strategy was implemented and three months after.

Before implementing the new marketing strategy, the company recorded the monthly sales data for a sample of 50 stores. After the strategy was launched and completed, they collected sales data for the same 50 stores. The marketing team hypothesizes that the new marketing strategy has led to an increase in sales. To test this hypothesis, they decide to use a two-sample right-tailed t-test.

You will perform the two-sample right-tailed t-test to determine if the mean sales after the implementation of the new marketing strategy are significantly higher than the mean sales before the strategy.

Step 1: State the Hypotheses

• H₀: (mean sales after the implementation of the new marketing strategy are equal to the mean sales before the implementation)
• H₁: (mean sales after the implementation of the new marketing strategy are greater than the mean sales before the implementation)

Step 2: Gather the Data

Sales data was randomly sampled from 50 stores for three months before and three months after implementing the new marketing strategy. The data is in the Excel file named sales_before_after_marketing_strategy.xlsx.

Step 3: Choose a Significance Level

Use a level of significance,

Step 4: Check for Equal Variances

You test the following in Excel.

• H₀: (variance in the sales data before and after the implementation of the new marketing strategy are equal)
• H₁: (variances of the sales data before and after the implementation of the new marketing strategy are not equal)

To perform the equality of variances test in Excel using the sales_before_after_marketing_strategy.xlsx file, perform the following.

1. Select F-Test Two Sample for Variances in the XL Miner Analysis Toolpak. The dialog box opens.



2. Place your cursor in the Variable 1 Range: box and select the data in column B.

3. Place your cursor in the Variable 2 Range: box and select the data in column C.

4. Place your cursor in the Output Range: box, select cell D2 in the Excel worksheet, and select OK.



From Excel, you obtain the following output. In cell E13, you can enter =2*E10 to obtain the 2-sided p-value for this test. This p-value (0.0005) is less than the level of significance of 0.05, so you can reject the null hypothesis and conclude that the variances among store sales before and after the implementation of the new marketing strategy are not equal.



For the two-sample hypothesis test, you will select the t-Test: Two-Sample Assuming Unequal Variances option in XL Miner.

Step 5: Perform a Statistical Test

Using the data in the sales_before_after_marketing_strategy.xlsx Excel file, you will now use the XL Miner Analysis Toolpak to find the p-value for this two-sample hypothesis test (right-tailed).

1. Select t-Test: Two-Sample Assuming Unequal Variance in the XL Miner Analysis Toolpak. The dialog box opens.



2. Place your cursor in the Variable 1 Range: box and select the data in column C.

3. Place your cursor in the Variable 2 Range: box and select the data in column B.

4. Enter 0 in the Hypothesized Mean Difference: box.

5. Place your cursor in the Output Range: box, select cell D16 in the Excel worksheet, and select OK.



Notice in the dialog box above that the column in the Variable 1 Range specifies the data related to the sales after the marketing strategy was implemented, After_Sales. The options you select for the Variable 1 and Variable 2 boxes in the t-test dialog box are crucial, because they directly determine the direction of your hypothesis test using the directional math operator in your alternative hypothesis.

The alternative hypothesis is:

H₁:

To set this alternative hypothesis up correctly in the t-test dialog box, perform the following:

• Variable 1 Range: Select the range for After_Sales (cells C2:C51).
• Variable 2 Range: Select the range for Before_Sales (cells B2:B51).

By correctly setting the Variable 1 and Variable 2 ranges, you ensure that the test is aligned with your alternative hypothesis and that the results will accurately reflect whether the sales after the marketing implementation are significantly greater than the sales before (Wright, n.d.).

The key here is to make sure that whatever group is on the left-hand side of the mathematical operator in the alternative hypothesis is the group that is specified in the Variable 1 Range box.

The Hypothesized Mean Difference: option is usually set to 0. This option directly relates to the null hypothesis.

The null hypothesis is:

H₀:

It means we start by assuming there is no difference between the sales before and after the marketing implementation. Another way to specify this is to say that the difference between the means of the two groups is zero. Subtract from both sides of the equation to obtain H₀: which we test against the alternative hypothesis that the sales after the marketing implementation are greater than the sales before.

You will obtain the following output in your Excel worksheet. If the test statistic (row labeled t Stat) is positive, then the row labeled P(T<=T) one-tail is the right-tailed p-value. The p-value for this two-sample hypothesis test is 0.0259.



Below is a table to help you calculate the p-value for a right-tailed test based on the sign of the test statistic using the P(T <= t) one-tail output from the XL Miner Analysis ToolPak (Wright, n.d.):

Test Statistic	P(T<=t) one tail	Right-Tailed p-value Calculation
Positive	p	p
Negative	p	1-p

• Positive Test Statistic: If the test statistic is positive, the P(T <= t) one-tail value is already the p-value for the right-tailed test.
• Negative Test Statistic: If the test statistic is negative, you need to subtract the P(T <= t) one-tail value from 1 to get the right-tailed p-value.

Step 6: Make a Decision

Since the p-value (level of significance), you reject the null hypothesis.

Step 7: Interpret the Results

Since the p-value (0.0259) is less than the significance level (0.05), you reject the null hypothesis. This means that there is sufficient evidence to conclude that the sales after the marketing implementation are significantly greater than the sales before the implementation.

The significant increase in sales after the implementation of the new marketing strategy validates its effectiveness. This suggests that the marketing strategy has positively impacted consumer behavior and sales performance. The marketing team at BrightFutures can be confident in their approach and consider it a successful initiative worth continuing or expanding to all stores.

Demonstrating a successful marketing strategy can strengthen BrightFuture’s position in the market, potentially attracting more customers and increasing market share. Effective marketing can enhance the brand image, making it more appealing to consumers and differentiating it from competitors.

Now that you have explored the practical implications of BrightFuture Electronics' marketing strategy using a two-sample hypothesis testing approach, it is time for you to apply your knowledge by performing a two-sample hypothesis test using another real-world scenario!

try it

For the scenario provided construct a two-sample t-test to address the business problem. Use a level of significance of 0.05. Ensure your solution contains:

1. A statement of the hypotheses.
2. A test for equality of variances.
3. An interpretation of what the result of the hypothesis test means in the context of the problem.

TrendyTech is a popular retail chain specializing in the latest tech gadgets and accessories. The company has two flagship stores: Store A, located in a bustling downtown area, and Store B, situated in a trendy suburban shopping center. Both stores have been performing well, but the management team wants to determine if Store A’s location gives it a significant sales advantage over Store B.

The management team has tasked you, the business data analyst, to compare the average monthly sales of Store A and Store B to see if Store A has significantly higher sales. This analysis will help in making strategic decisions about future store locations and marketing investments.

The business data analytics team collects monthly sales data for the past year from both stores, resulting in 50 observations for each store. This data is in the Excel file named store_sales_comparison.xlsx.

Solution:

The hypotheses are:

• H₀: (mean monthly sales of Store A are equal to the mean monthly sales of Store B)
• H₁: (mean monthly sales of Store A are greater than or equal to the mean monthly sales of Store B)

You need to perform the equality of variances test to determine which t-test you will select for the two-sample hypothesis test.

The hypotheses are:

• (variance in monthly sales of Store A is equal to the variance in monthly sales of Store B)
• (variance in monthly sales of Store A is not equal to the variance in monthly sales of Store B)

Use the F-Test Two-Sample for Variances option in the XL Miner Analysis Toolpak. Your options should look like:



From Excel, you obtain the following output. In cell E13, you can enter =2*E10 to obtain the 2-sided p-value for this test. This p-value is less than 0.0001 so you can reject the null hypothesis and conclude that the variances among the monthly stores of both stores are not equal.



For the two-sample hypothesis test, you will select the t-Test: Two-Sample Assuming Unequal Variances option in XL Miner.



You obtain the following output from Excel. If the test statistic (row labeled t Stat) is positive, then the row labeled P(T<=T) one-tail is the right-tailed p-value. The p-value for this two-sample hypothesis test is 0.1413.



Since the p-value (0.1413) is greater than the significance level (0.05), you fail to reject the null hypothesis. This means there is not enough evidence to conclude that the mean monthly sales of Store A are greater than the mean monthly sales of Store B.

watch

Check out this video on conducting a two-sample t-test to analyze average sales.

think about it

In the previous Try It exercise, you failed to reject the null hypothesis and concluded that there was not enough evidence to conclude that the mean monthly sales of Store A are greater than the mean monthly sales of Store B.

What are the practical implications for TrendyTech based on the results of this hypothesis test?

Location Strategy: The lack of significant difference in sales between Store A and Store B suggests that the downtown location does not provide a substantial sales advantage over the suburban location. This insight can guide future decisions on store placements, indicating that suburban areas might be just as viable as downtown locations.

Marketing Investments: Since there is no evidence to support that Store A is performing better than Store B in terms of sales, TrendyTech might consider distributing marketing resources more evenly between the two locations. Alternatively, they could focus on other factors that might drive sales, such as in-store promotions, customer service enhancements, or product variety.

REFERENCES

Wright, D. (n.d.). Tail of the test: Interpreting Excel data analysis t-test output. www.drdawnwright.com/tail-of-the-test-interpreting-excel-data-analysis-t-test-output/.