Question

In the paper "Happiness for Sale: Do Experiential Purchases Make Consumers Happier than Material Purchases?" (Journal of Consumer Research [2009]: \(188-197\) ), the authors distinguish between spending money on experiences (such as travel) and spending money on material possessions (such as a car). In an experiment to determine if the type of purchase affects how happy people are after the purchase has been made, 185 college students were randomly assigned to one of two groups. The students in the "experiential" group were asked to recall a time when they spent about \(\$ 300\) on an experience. They rated this purchase on three different happiness scales that were then combined into an overall measure of happiness. The students assigned to the "material" group recalled a time that they spent about \(\$ 300\) on an object and rated this purchase in the same manner. The mean happiness score was 5.75 for the experiential group and 5.27 for the material group. Standard deviations and sample sizes were not given in the paper, but for purposes of this exercise, suppose that they were as follows: \begin{tabular}{|ll|} \hline Experiential & Material \\ \hline\(n_{1}=92\) & \(n_{2}=93\) \\ \(s_{1}=1.2\) & \(s_{2}=1.5\) \\ \hline \end{tabular} Using the following Minitab output, carry out a hypothesis test to determine if these data support the authors' conclusion that, on average, "experiential purchases induced more reported happiness." Use \(\alpha=0.05\) Two-Sample T-Test and Cl Sample \(\begin{array}{rrrrr}\text { ple } & \text { N } & \text { Mean } & \text { StDev } & \text { SE Mean } \\ 1 & 92 & 5.75 & 1.20 & 0.13 \\ 2 & 93 & 5.27 & 1.50 & 0.16\end{array}\) Difference \(=\operatorname{mu}(1)-\operatorname{mu}(2)\) Estimate for difference: 0.480000 \(95 \%\) lower bound for difference: 0.149917 T-Test of difference \(=0(\mathrm{vs}>): \mathrm{T}\) -Value \(=2.40 \mathrm{P}\) -Value \(=\) \(0.009 \mathrm{DF}=175\)

Short answer

The statistical test supports the claim that, on average, 'experiential purchases induced more reported happiness.' The test provides enough evidence to reject the null hypothesis and conclude that the mean of reported happiness is greater for the experiential group compared to the material one.

Step-by-step solution

Setup Hypotheses

Firstly, it's important to define the null and alternative hypotheses. The null hypothesis (\(H_0\)) is that there is no difference in the average happiness level between the experiential purchase group and the material purchase group. The alternative hypothesis (\(H_a\)) is that there is a difference. These can be written as follows:\n\nNull hypothesis (\(H_0\)): \(\mu_1 = \mu_2\)\n, Alternate hypothesis (\(H_a\)): \(\mu_1 > \mu_2\)

Test calculation

The calculated test statistic from the Minitab output is 2.40 and p-value is 0.009.

Decision making

Now compare the p-value with the level of significance (Alpha). If the p-value is less than the significance level (\(\alpha = 0.05\)), reject the null hypothesis. Since p-value (0.009) < significance level (0.05) We reject \(H_0\).

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is foundational to any hypothesis testing procedure. In the context of our exercise, the null hypothesis (\(H_0\)) posits that there is no statistical difference in happiness levels between groups making experiential purchases and those making material purchases. Formally, it states that \(\mu_1 = \mu_2\), where \(\mu_1\) and \(\mu_2\) represent the mean happiness scores for the experiential and material groups, respectively.

The alternative hypothesis (\(H_a\) or \(H_1\)), on the other hand, challenges the null by proposing that a specific difference exists; in our case, that experiential purchases lead to higher happiness, formally stated as \(\mu_1 > \mu_2\). It guides the direction of the statistical test, indicating that we're specifically looking for evidence of greater happiness from experiential spending.

P-value

The p-value plays a critical role in hypothesis testing. It quantifies the probability of observing a test statistic as extreme as, or more extreme than, the value observed in your study, assuming that the null hypothesis is true. In simpler terms, it's a measure of how surprising your data is under the assumption that there are no real effects or differences ('surprising' generally means how well your data supports the null hypothesis).

For our example, the p-value is 0.009. This number is a probability, falling between 0 and 1, and it tells us how compatible our data are with the null hypothesis. A very small p-value, such as ours, indicates that if the null hypothesis were true, it would be very unlikely to observe the data we have. This leads to stronger evidence against the null hypothesis, suggesting that our alternative hypothesis might be the correct one.

Statistical Significance

Statistical significance is the determination of whether or not the results of a statistical test, such as our t-test, supports the rejection of the null hypothesis. This decision is made by comparing the p-value to the chosen significance level, often denoted by \(\alpha\). The significance level represents the probability of committing a Type I error — rejecting a true null hypothesis.

In hypothesis testing, if the p-value is less than the chosen \(\alpha\) level (usually 0.05), the results are deemed statistically significant. This implies that the evidence in the sample is strong enough to conclude that there is a statistically detectable effect. In the given exercise, with a p-value of 0.009 and an \(\alpha\) of 0.05, the result is statistically significant, and we would reject the null hypothesis in favor of the alternative.

Two-Sample t-Test

The two-sample t-test is a statistical test used to compare the means of two independent groups. It offers a way to evaluate whether the difference in means is statistically significant or if it likely occurred by random chance. The t-test takes into account both the sample sizes and variability within the sample data of the two groups, which is often reflected in the standard deviation values.

Applying this to our situation, the two-sample t-test is used to contrast the happiness scores of the experiential and material purchase groups. We are provided Minitab output data showing a calculated test statistic (also called t-value) and an associated p-value. With a t-value of 2.40 and our previously examined low p-value, the test indicates that we have sufficient evidence to believe that there is a significant difference in happiness scores, favoring the experiential group.