Question

The international polling organization Ipsos reported data from a survey of 2000 randomly selected Canadians who carry debit cards (Canadian Account Habits Survey, July 24, 2006). Participants in this survey were. asked what they considered the minimum purchase amount for which it would be acceptable to use a debit card. Suppose that the sample mean and standard deviation were \(\$ 9.15\) and \(\$ 7.60\) respectively. (These values are consistent with a histogram of the sample data that appears in the report.) Do these data provide convincing evidence that the mean minimum purchase amount for which Canadians consider the use of a debit card to be appropriate is less than \(\$ 10\) ? Carry out a hypothesis test with a significance level of \(.01\).

Short answer

The final decision will be based on the calculated Z-score. If the Z-score is less than -2.33, then the null hypothesis is rejected and there is evidence to suggest that the mean minimum purchase amount for which Canadians consider the use of a debit card to be appropriate is less than \$10. If the Z-score is not less than -2.33, then the null hypothesis is not rejected.

Step-by-step solution

Formulate the null and alternate hypotheses

The null hypothesis \(H_0\) is that the mean minimum purchase amount for which Canadians consider the use of a debit card to be appropriate is \$10. So \(H_0 : \mu = 10\). The alternate hypothesis \(H_1\) is that the mean minimum purchase amount for which Canadians consider the use of a debit card to be appropriate is less than \$10. So \(H_1 : \mu < 10\).

Compute the test statistic

The sample mean is \$9.15 and the sample standard deviation is \$7.60. The sample size is 2000. The test statistic \(Z\) is calculated as follows: \(Z = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}} = \frac{9.15 - 10}{7.60/\sqrt{2000}}\).

Determining the rejection region

This is a left-tailed test since the alternative hypothesis is \(\mu < 10\). The significance level for the test is .01. The Z-value that corresponds to this level of significance for a left-tailed test is approximately -2.33. This is the critical value. If our calculated Z score from Step 2 is less than this critical value, we will reject the null hypothesis.

Decision on the hypothesis

If the test statistic is less than the critical value (-2.33), then we reject the null hypothesis in favor of the alternate hypothesis. This would mean there is sufficient evidence to suggest that the mean minimum purchase amount for which Canadians consider the use of a debit card to be appropriate is less than \$10. If the test statistic is not less than the critical value, then we fail to reject the null hypothesis.

Interpret the result

After comparing the test statistic with the critical value and making the decision about the hypothesis, we interpret the result. If we reject the null hypothesis, then it means there is sufficient evidence to suggest that Canadians consider the use of a debit card to be appropriate for purchases of less than \$10. If we fail to reject the null hypothesis, then there isn't enough evidence to support this.

Key concepts

Null and Alternate Hypotheses

Understanding the role of null and alternate hypotheses is crucial when conducting hypothesis testing in statistics. These hypotheses are the core of any statistical test and set the stage for the analysis.

The null hypothesis, denoted as \(H_0\), represents a statement of no effect or no difference and is the assumption that we initially consider to be true. In the context of the Ipsos survey example, the null hypothesis posits that the mean minimum purchase amount at which it is acceptable to use a debit card among Canadians is \(\$10\), or \(H_0 : \mu = 10\).

The alternate hypothesis, denoted as \(H_1\) or \(H_a\), is a statement that contradicts the null hypothesis. It is what the researcher aims to support. For our debit card usage example, the alternate hypothesis is that the average minimum purchase amount is less than \(\$10\), formulated as \(H_1 : \mu < 10\).

The establishment of these hypotheses lays the foundation for the subsequent steps in hypothesis testing which involve calculating the test statistic, evaluating significance, and making a decision regarding which hypothesis is supported by the sample data.

Test Statistic Calculation

The test statistic is a standardized value that measures the degree to which the sample data diverge from the null hypothesis. It's a pivotal part of the hypothesis testing procedure because it enables comparison to the distribution under the null hypothesis.

In our example, the test statistic is calculated using the formula: \[ Z = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}} \] where \(\bar{X}\) is the sample mean, \(\mu_0\) is the mean under the null hypothesis, \(\sigma\) is the sample standard deviation, and \(n\) is the sample size. Plugging in the values from the Ipsos survey, we can calculate the Z score, which you compare with a critical value to make a decision about the hypothesis.

It's important for students to familiarize themselves with the formula and understand that the test statistic not only takes into account the difference between the sample mean and the hypothesized population mean but also considers the sample size and variability in the data. A larger test statistic generally indicates a greater degree of divergence between the sample data and the null hypothesis.

Significance Level

The significance level, denoted by \(\alpha\), is a threshold set by the researcher to decide whether to reject the null hypothesis. It's the probability of making the mistake of rejecting the null hypothesis when it is actually true – this is known as a Type I error.

In hypothesis testing, common choices for \(\alpha\) are 0.05, 0.01, or 0.10, depending on how strict the researcher wants to be. A smaller \(\alpha\) suggests a more rigorous test. For instance, in the given exercise, a significance level of 0.01 was chosen, indicating that there is only a 1% chance the researcher is willing to take to incorrectly reject the null hypothesis when it's true. This implies that the evidence must be quite strong to support the claim that the mean minimum purchase amount is indeed less than \(\$10\).

Deciding on the significance level before collecting data is imperative because it informs the rest of the hypothesis testing process, particularly in determining the critical value and interpreting the p-value.

Critical Value Analysis

The critical value is a point on the test distribution that is compared to the test statistic to determine whether to reject the null hypothesis. It serves as a boundary: if the test statistic crosses this boundary, the result is statistically significant.

For a given significance level, the critical value can be found using statistical tables (like the Z-table for normal distribution). In our example, we have a left-tailed test, and the critical value associated with an \(\alpha\) of 0.01 is approximately -2.33. This means that if the calculated Z score is less than -2.33, the corresponding p-value will be smaller than 0.01, leading us to reject the null hypothesis.

It is vital for students to grasp that critical value analysis is the process of comparing the calculated test statistic to this predefined threshold. Understanding this concept will enable them to correctly interpret the outcome of the hypothesis test and decide whether their data provide convincing evidence against the null hypothesis, as was sought in the Ipsos survey example to determine Canadians' debit card use preferences.