Answer :
Let's go through each part of the problem step-by-step:
### Part (a): Examine Each Group's Prices Graphically and Assess if [tex]$t$[/tex] Procedures Are Appropriate
To determine if [tex]$t$[/tex] procedures are appropriate, examine the graphical representation of the data through histograms or box plots. The [tex]$t$[/tex] procedures generally assume:
1. The samples are independent.
2. The data is approximately normally distributed (especially for small sample sizes).
Given the data:
1. Neutral Group: [tex]\[0.00, 2.00, 0.00, 1.00, 0.50, 0.00, 0.50, 2.00, 1.00, 0.00, 0.00, 0.00, 0.00, 1.00\][/tex]
2. Sadness Group: [tex]\[3.00, 4.00, 0.50, 1.00, 2.50, 2.00, 1.50, 0.00, 1.00, 1.50, 1.50, 2.50, 4.00, 3.00, 3.50, 1.00\][/tex]
Review the histograms or box plots for symmetry and outliers:
- The Neutral Group shows a cluster of zeros, suggesting potential skewness.
- The Sadness Group shows a more even spread, potentially closer to normality.
Given the sample size (14 in the neutral group and 16 in the sadness group), the central limit theorem may somewhat alleviate normality concerns, indicating [tex]$t$[/tex] procedures can be reasonably used.
### Part (b): Table of Sample Size, Mean, and Standard Deviation
The statistics for each group can be summarized as follows:
| Group | Sample Size (n) | Mean ([tex]$\bar{x}$[/tex]) | Standard Deviation (s) |
|---------|-----------------|------------------|------------------------|
| Neutral | 14 | 0.571 | 0.730 |
| Sadness | 16 | 2.031 | 1.231 |
### Part (c): Null and Alternative Hypotheses
To compare the mean purchase prices between the two groups, we establish the following hypotheses:
- Null Hypothesis ([tex]$H_0$[/tex]): There is no difference in mean purchase prices between the two groups. [tex]\( \mu_{\text{neutral}} = \mu_{\text{sadness}} \)[/tex]
- Alternative Hypothesis ([tex]$H_1$[/tex]): There is a difference in mean purchase prices between the two groups. [tex]\( \mu_{\text{neutral}} \neq \mu_{\text{sadness}} \)[/tex]
### Part (d): Perform the Significance Test
We conduct a two-sample [tex]$t$[/tex]-test, assuming unequal variances:
- Test Statistic (t-statistic): -3.876
- Degrees of Freedom (df): 28
- P-Value: 0.000586
With the significance level [tex]$\alpha = 0.05$[/tex]:
- Given the p-value (0.000586) is much less than [tex]$\alpha$[/tex], we reject the null hypothesis.
Conclusion: There is sufficient evidence at the 0.05 significance level to conclude that there is a statistically significant difference in mean purchase prices between the neutral group and the sadness group.
### Part (e): Confidence Interval for the Mean Difference
We construct a 95% confidence interval for the difference in mean purchase prices:
- Mean Difference ([tex]$\bar{x}_{\text{sadness}} - \bar{x}_{\text{neutral}}$[/tex]): [tex]\(2.031 - 0.571 \approx 1.460\)[/tex]
- Standard Error of the Difference (se_diff): Calculated and used in the confidence interval formula.
- 95% Confidence Interval: (0.746, 2.174)
Interpretation: We are 95% confident that the true difference in mean purchase prices between the sadness and neutral groups lies between [tex]$0.746 and $[/tex]2.174. This reinforces our earlier finding from the hypothesis test, suggesting a meaningful difference in spending behavior between the two groups.
### Part (a): Examine Each Group's Prices Graphically and Assess if [tex]$t$[/tex] Procedures Are Appropriate
To determine if [tex]$t$[/tex] procedures are appropriate, examine the graphical representation of the data through histograms or box plots. The [tex]$t$[/tex] procedures generally assume:
1. The samples are independent.
2. The data is approximately normally distributed (especially for small sample sizes).
Given the data:
1. Neutral Group: [tex]\[0.00, 2.00, 0.00, 1.00, 0.50, 0.00, 0.50, 2.00, 1.00, 0.00, 0.00, 0.00, 0.00, 1.00\][/tex]
2. Sadness Group: [tex]\[3.00, 4.00, 0.50, 1.00, 2.50, 2.00, 1.50, 0.00, 1.00, 1.50, 1.50, 2.50, 4.00, 3.00, 3.50, 1.00\][/tex]
Review the histograms or box plots for symmetry and outliers:
- The Neutral Group shows a cluster of zeros, suggesting potential skewness.
- The Sadness Group shows a more even spread, potentially closer to normality.
Given the sample size (14 in the neutral group and 16 in the sadness group), the central limit theorem may somewhat alleviate normality concerns, indicating [tex]$t$[/tex] procedures can be reasonably used.
### Part (b): Table of Sample Size, Mean, and Standard Deviation
The statistics for each group can be summarized as follows:
| Group | Sample Size (n) | Mean ([tex]$\bar{x}$[/tex]) | Standard Deviation (s) |
|---------|-----------------|------------------|------------------------|
| Neutral | 14 | 0.571 | 0.730 |
| Sadness | 16 | 2.031 | 1.231 |
### Part (c): Null and Alternative Hypotheses
To compare the mean purchase prices between the two groups, we establish the following hypotheses:
- Null Hypothesis ([tex]$H_0$[/tex]): There is no difference in mean purchase prices between the two groups. [tex]\( \mu_{\text{neutral}} = \mu_{\text{sadness}} \)[/tex]
- Alternative Hypothesis ([tex]$H_1$[/tex]): There is a difference in mean purchase prices between the two groups. [tex]\( \mu_{\text{neutral}} \neq \mu_{\text{sadness}} \)[/tex]
### Part (d): Perform the Significance Test
We conduct a two-sample [tex]$t$[/tex]-test, assuming unequal variances:
- Test Statistic (t-statistic): -3.876
- Degrees of Freedom (df): 28
- P-Value: 0.000586
With the significance level [tex]$\alpha = 0.05$[/tex]:
- Given the p-value (0.000586) is much less than [tex]$\alpha$[/tex], we reject the null hypothesis.
Conclusion: There is sufficient evidence at the 0.05 significance level to conclude that there is a statistically significant difference in mean purchase prices between the neutral group and the sadness group.
### Part (e): Confidence Interval for the Mean Difference
We construct a 95% confidence interval for the difference in mean purchase prices:
- Mean Difference ([tex]$\bar{x}_{\text{sadness}} - \bar{x}_{\text{neutral}}$[/tex]): [tex]\(2.031 - 0.571 \approx 1.460\)[/tex]
- Standard Error of the Difference (se_diff): Calculated and used in the confidence interval formula.
- 95% Confidence Interval: (0.746, 2.174)
Interpretation: We are 95% confident that the true difference in mean purchase prices between the sadness and neutral groups lies between [tex]$0.746 and $[/tex]2.174. This reinforces our earlier finding from the hypothesis test, suggesting a meaningful difference in spending behavior between the two groups.
