91Ó°ÊÓ

In hypothesis testing, especially with small sample sizes (typically under 30), we often rely on the t-distribution rather than the normal distribution. This is because the t-distribution takes into account the increased variability found in smaller samples, providing a more accurate result.
The t-distribution is similar in shape to the normal distribution but has heavier tails. It accounts for the uncertainty in the estimate of the standard deviation when working with small samples. As sample size increases, the t-distribution approaches the normal distribution.
A key component of using the t-distribution is understanding the degrees of freedom, which typically equals the sample size minus one \(df = n - 1\). In our wedding cost sample, with 8 weddings, the degrees of freedom is 7. This affects the critical value we use in our hypothesis test.

The sample mean is crucial in hypothesis testing as it is our best estimate of the population mean. It provides a central value around which the data points are distributed.
To calculate the sample mean, sum all the data values and divide by the number of observations. In our Caribbean weddings example, the total of the sample data equates to 79.3 when added together. By dividing by the number of weddings (8), we get a sample mean \( \bar{x} = 9.9125 \).
The sample mean serves as the starting point in hypothesis tests, evaluating differences between the sample mean and the specified value (here, $10,000), thereby assisting in making inferences about the population mean.

The significance level, often denoted by \( \alpha \), is the probability of rejecting the null hypothesis when it is true. It represents the risk we are willing to take for a type I error. Common significance levels include 0.05, 0.01, and 0.10.
In our test about the cost of Caribbean weddings, we use a significance level of 0.05. This implies there is a 5% risk of concluding that the mean wedding cost is less than \(10,000 when it might actually not be. At this level, the decision rule involves comparing the calculated t-statistic to the critical t-value found in the t-distribution table with 7 degrees of freedom.
This critical value acts as a threshold determining if the observed data sufficiently supports rejecting the null hypothesis. Since the calculated t-statistic was not less than the critical t-value, we did not find enough evidence to support the alternative hypothesis that the cost is below \)10,000.

A one-tailed test in hypothesis testing checks whether a sample mean is either greater than or less than a known value. It provides insight into the question of a direction-specific hypothesis, different from a two-tailed test which would check for any significant difference irrespective of direction.
For the Caribbean wedding cost exercise, the alternative hypothesis \( H_a: \mu < 10\) specifies a "less than" test, making it one-tailed. We're solely interested in whether costs are significantly lower than $10,000.
In practical terms, this affects how we interpret the t-statistic and the critical value from the t-distribution chart. A one-tailed test at a 0.05 level leads to the critical value being more extreme (further from zero) compared to a two-tailed test, suggesting that the evidence needed to reject the null must be stronger. Thus, using a one-tailed test helps refine our conclusions about specific directions in hypothesis testing.