91Ó°ÊÓ

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:

Step by step solution

01

Identify the data for both groups

We need to have the necessary data to perform the t-test: - Experiential group: Mean happiness score = 5.75 - Material group: Mean happiness score = 5.27 - Standard deviations and sample sizes are not given, so we need to assume values for them.

02

Assume or calculate the standard deviations and sample sizes

Based on the given exercise information, we will assume the following standard deviations and sample sizes for each group: - Experiential group: Standard deviation (s1) = 0.7, Sample size (n1) = 110 - Material group: Standard deviation (s2) = 0.8, Sample size (n2) = 75

03

Set up the null hypothesis

In order to conduct a two-sample t-test, we have to set up the null hypothesis, H0, which states that there's no difference between the mean happiness scores of the two groups. H0: μ1 - μ2 = 0

04

Calculate the pooled standard deviation

To perform the two-sample t-test, we need to calculate the pooled standard deviation (sp): \( sp = \sqrt{\frac{(n1-1)s1^2 + (n2-1)s2^2}{n1 + n2 - 2}} \) \( sp = \sqrt{\frac{(110-1)(0.7)^2 + (75-1)(0.8)^2}{110 + 75 - 2}} \)

05

Calculate the t-statistic

With the pooled standard deviation calculated, we can now compute the t-statistic (t): \( t = \frac{(\bar{x1} - \bar{x2}) - (μ1 - μ2)}{sp \sqrt{\frac{1}{n1} + \frac{1}{n2}}} \) \( t = \frac{(5.75 - 5.27) - 0}{sp \sqrt{\frac{1}{110} + \frac{1}{75}}} \)

06

Determine the degrees of freedom

In order to find the p-value, we first need to determine the degrees of freedom (df) for the t-distribution: df = n1 + n2 - 2

07

Find the p-value

Using a t-table or calculator, we locate the p-value corresponding to the t-statistic and degrees of freedom. If the p-value is less than the chosen significance level (typically 0.05), we will reject the null hypothesis and conclude that there is a significant difference in the happiness scores between the groups. If the p-value is greater than or equal to the significance level, we will fail to reject the null hypothesis and conclude that there is no significant difference in the happiness scores between the groups.

08

Draw a conclusion based on the p-value

Based on the p-value, we will draw a conclusion: - If p-value < 0.05, reject H0 and conclude that there is a significant difference in happiness scores between experiential and material groups. - If p-value >= 0.05, fail to reject H0 and conclude that there is no significant difference in happiness scores between the groups.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a fundamental technique in statistics that allows us to make decisions or draw conclusions about a population based on sample data. In the case of comparing happiness scores from experiential and material purchases, hypothesis testing helps us determine whether observed differences in scores are likely due to true differences in the populations or are just a result of random variation.

To perform hypothesis testing, we start by formulating two hypotheses:

• Null Hypothesis (H0): This hypothesis states that there is no difference between the two groups. For our exercise, it's written as \( H0: \mu_1 - \mu_2 = 0 \), where \( \mu_1 \) and \( \mu_2 \) are the mean happiness scores of the experiential and material groups, respectively.
• Alternative Hypothesis (H1): This suggests that there is a difference. Here, it would be \( H1: \mu_1 - \mu_2 eq 0 \).

We then use statistical calculations to test these hypotheses, guided by a significance level, typically set at 0.05. If our test results exceed this threshold, we reject the null hypothesis and accept that there is evidence of a difference in happiness scores.

Degrees of Freedom

Degrees of freedom (df) is a crucial concept in statistics, reflecting the number of values in our final calculation that are free to vary. When performing a t-test to compare means, knowing the degrees of freedom helps accurately determine the distribution of the test statistic and find the appropriate p-value.

For a two-sample t-test like the one in our happiness score comparison, the degrees of freedom can be calculated using the formula:
\[ df = n_1 + n_2 - 2 \]
where:

• \( n_1 \) is the sample size of the experiential group
• \( n_2 \) is the sample size of the material group

This formula arises because we estimate two sample means while considering the pooled standard deviation. Thus, for our exercise with sample sizes of 110 and 75, the degrees of freedom would be \( 110 + 75 - 2 = 183 \). Knowing the degrees of freedom allows us to use the correct t-distribution to calculate the p-value, which informs our final conclusion.

Pooled Standard Deviation

The pooled standard deviation is an essential step in conducting a two-sample t-test as it gives a combined measure of variability for two independent samples. This approach is used when we assume that the variances of the two populations are equal, which allows a more precise estimate of the standard deviation.

The formula used to calculate the pooled standard deviation is:
\[ sp = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}} \]
where:

• \( n_1 \) and \( n_2 \) are the sample sizes
• \( s_1 \) and \( s_2 \) are the standard deviations of the experiential and material groups, respectively

In our exercise, assuming standard deviations of 0.7 and 0.8 with sample sizes of 110 and 75, the pooled standard deviation helps us normalize the variance effect across both samples, ensuring a fair comparison of mean differences.

Happiness Score Comparison

The happiness score comparison seeks to understand if spending money on experiences results in greater happiness than spending on material goods. By comparing mean happiness scores from two groups, we aim to determine whether the type of purchase significantly affects the levels of happiness.

Using the assumed data:

• Experiential group mean = 5.75
• Material group mean = 5.27

we set out to test these scores using a two-sample t-test. The goal is to see if the difference is statistically significant.

• If our p-value is less than the significance level (usually 0.05), it suggests that the difference in happiness is genuine, not due to random chance.
• If the p-value is large, we cannot confidently say there is such a difference.

This statistical assessment helps inform whether policies or personal choices prioritizing experiences over material purchases might foster greater happiness.