The Cost Of Weddings In The United States Has Skyrocketed

The Cost Of Weddings In The United States Has Skyrocketed In Recent characters

The cost of weddings in the United States has skyrocketed in recent years. As a result, many couples are opting to have their weddings in the Caribbean. A Caribbean vacation resort recently advertised in Bride Magazine that the cost of a Caribbean wedding was less than $10,000. Listed below is a total cost for a sample of 8 Caribbean weddings: $9,700, $9,400, $11,700, $9,000, $9,100, $10,500, $9,100, $9,800.

The assignment involves conducting a hypothesis test to determine whether the mean cost of Caribbean weddings is less than $10,000 based on the sample data provided.

Paper For Above instruction

Introduction

Wedding costs have been on the rise in the United States, prompting many to consider destination weddings in popular locations such as the Caribbean. Given the advertised claim that Caribbean weddings cost less than $10,000, it is pertinent to statistically evaluate this assertion based on sample data. Statistical hypothesis testing provides a framework to assess whether the sample data supports the claim that the average wedding cost in the Caribbean is below this threshold or not.

Formulating the Hypotheses

The null hypothesis (H₀) represents the assumption that there is no difference from the claimed average of $10,000, i.e., that the true mean cost μ is equal to $10,000. The alternative hypothesis (H₁) corresponds to the claim that the mean cost is less than $10,000. Mathematically, these are expressed as:

• H₀: μ = 10,000
• H₁: μ

These hypotheses are suitable for a one-tailed t-test, considering the directional nature of the claim.

Critical Value Rule at 5% Significance Level

At a significance level (α) of 0.05 and with a small sample size (n=8), the critical value for a one-tailed t-test is determined from the t-distribution table. Degrees of freedom (df) = n - 1 = 7. The critical t-value for a left-tailed test at α = 0.05 and df = 7 is approximately -1.895. Therefore, the decision rule is:

If the calculated t-statistic

Calculating the Test Statistic

Using the sample data: 9,700; 9,400; 11,700; 9,000; 9,100; 10,500; 9,100; 9,800, we first compute the sample mean and sample standard deviation.

Sum of the data points: 9,700 + 9,400 + 11,700 + 9,000 + 9,100 + 10,500 + 9,100 + 9,800 = 78,400

Sample mean: 78,400 / 8 = 9,800

Next, calculate the sample standard deviation (s):

s = √[Σ(x_i - x̄)² / (n - 1)]

Calculating the squared deviations:

• (9,700 - 9,800)² = (−100)² = 10,000
• (9,400 - 9,800)² = (−400)² = 160,000
• (11,700 - 9,800)² = 1,900² = 3,610,000
• (9,000 - 9,800)² = (−800)² = 640,000
• (9,100 - 9,800)² = (−700)² = 490,000
• (10,500 - 9,800)² = 700² = 490,000
• (9,100 - 9,800)² = (−700)² = 490,000
• (9,800 - 9,800)² = 0

Sum of squared deviations: 10,000 + 160,000 + 3,610,000 + 640,000 + 490,000 + 490,000 + 490,000 + 0 = 6,890,000

Sample variance: 6,890,000 / (8 - 1) = 6,890,000 / 7 ≈ 985,714.29

Sample standard deviation: √985,714.29 ≈ 992.83

Calculating the t-statistic:

t = (x̄ - μ₀) / (s / √n) = (9,800 - 10,000) / (992.83 / √8) ≈ (−200) / (992.83 / 2.828) ≈ (−200) / 351.18 ≈ -0.569

Decision Regarding the Null Hypothesis

The calculated t is approximately -0.569, which is not less than the critical value of -1.895. Therefore, we fail to reject the null hypothesis at the 5% significance level.

Conclusion

Based on the sample data and the hypothesis test conducted, there is insufficient evidence at the 5% level to support the claim that the mean cost of Caribbean weddings is less than $10,000. In other words, the data does not provide convincing statistical support to confirm the advertised lower cost.

Implications and Error Considerations

Considering the results, the type of error that could have been committed is a Type II error. This occurs when we fail to reject a false null hypothesis. In this context, it means that in reality, the mean wedding cost might be less than $10,000, but our sample data and test did not reflect this sufficiently to reject H₀.

The potential ramification of committing a Type II error is that consumers or industry analyses might underestimate the affordability of Caribbean weddings. Conversely, if the null hypothesis were incorrectly rejected (Type I error), it might falsely suggest that the average wedding cost is below $10,000, possibly influencing consumer decisions or marketing strategies based on inaccurate conclusions.

In the real-world scenario, understanding the balance between these errors is critical. A Type II error may lead to missed opportunities for marketing or customer planning, while a Type I error could lead to overestimating cost savings, influencing business strategies and consumer expectations.

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