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Regression analysis is a powerful statistical method used to examine the relationship between two or more variables. In our exercise, we focus on simple linear regression, which helps us determine how a change in one variable impacts another. The goal is to find the best-fitting line through the data points. This line is represented by the regression equation

• \( \hat{y} = 1.85 + 0.08x \)

where \( \hat{y} \) is the predicted value, \( 1.85 \) is the intercept, and \( 0.08 \) is the slope. The slope indicates how much \( y \) is expected to increase when \( x \) increases by one unit.
The hypothesis test in our exercise aims to find out if this slope is significantly different from zero. If it is, it suggests a meaningful relationship between the variables.

The t-distribution is a type of probability distribution that is bell-shaped and symmetrical, similar to the normal distribution but with thicker tails. It is particularly useful for small sample sizes or when the population standard deviation is unknown.
In our exercise, we use the t-distribution to conduct hypothesis testing for the slope of the regression line.

• The test statistic calculated is \( t \approx 2.67 \)
• The sample size minus 2 gives us the degrees of freedom, \( df = 10 \).

With this, we look up critical values in a t-distribution table, using the degrees of freedom and significance level to determine how extreme our test result is relative to what we would expect by chance.

The significance level, often denoted as \( \alpha \), is the probability of rejecting the null hypothesis when it is actually true. In simpler terms, it represents the threshold for determining whether an observed effect is statistically significant.
In our situation, we use a significance level of \( 0.05 \). This means that we're willing to accept a 5% chance of incorrectly rejecting the null hypothesis, which states that the slope of the regression line is zero. Choosing the right significance level is crucial, as smaller values imply requiring stronger evidence to reject the null hypothesis.

• It balances between Type I error (false positive) and statistical power.
• A conventional choice like 0.05 helps in conveying consistent scientific knowledge.

The critical value is the threshold that a test statistic must exceed for us to reject the null hypothesis. It acts as a boundary between the region where you could confidently accept or reject the null hypothesis in a hypothesis test.
In our exercise, we look up the critical value for the t-distribution:

• With \( 10 \) degrees of freedom
• A two-tailed test at a \( 0.05 \) significance level
• Our critical t-value is \( \pm 2.228 \)

This means that our test statistic, \( 2.67 \), being greater than 2.228, falls in the critical region where we can reject the null hypothesis.
Thus, the critical value acts as a decisive tool for making a valid conclusion from our statistical test.