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A t-test is a statistical method used to determine if there is a significant difference between the means of two groups. In this exercise, we have a sample of people who quit smoking, and we wish to know if their average weight change is significantly positive. This is where the t-test comes in handy. It helps us to check if the observed mean weight change could have happened by chance. The formula for the t-test is: \[ t = \frac{\bar{X} - \mu}{s / \sqrt{n}} \]where:

• \(\bar{X}\) is the sample mean,
• \(\mu\) is the population mean under the null hypothesis,
• \(s\) is the sample standard deviation,
• \(n\) is the sample size.

This formula helps us to calculate the t-score, which can be compared to a critical value from the t-distribution to decide if the null hypothesis should be rejected.

Statistical significance plays a crucial role in hypothesis testing. It tells us whether our results from a study are likely to have occurred by chance. In this context, the **p-value** is used to determine the statistical significance. For a result to be statistically significant, the p-value must be less than the chosen significance level, \(\alpha\). In this exercise, \(\alpha\) is set at 0.05, which means there is a 5% risk of concluding that an effect exists when there is none. If the computed t-score leads to a p-value lower than 0.05, the results are considered statistically significant. This tells us that the observed mean weight change isn't likely due to random variations alone.

The null hypothesis (\(H_0\)) is a statement used in hypothesis testing that assumes there is no effect or no difference. It serves as the default or starting assumption. In our exercise, the null hypothesis is that the true mean change in weight of people who quit smoking is not positive. Mathematically, this is represented as:\(H_0: \mu \leq 0\). We use statistical tests, like the t-test, to find evidence against it. Rejecting the null hypothesis suggests that there is enough evidence to support the idea that the true mean weight change is positive. If the null hypothesis is not rejected, the evidence is insufficient to suggest a difference from zero.

An alternative hypothesis (\(H_a\)) is the statement that contradicts the null hypothesis. It proposes that there is an effect or a difference. In this context, it asserts that the true mean weight change is indeed positive after quitting smoking. Mathematically, it is:\(H_a: \mu > 0\).A successful hypothesis test will reject the null hypothesis, leading us to accept the alternative hypothesis as the more likely scenario.In the example from the exercise, by conducting our t-test and finding that the test statistic falls beyond the critical value, we conclude that there is significant statistical evidence in favor of the alternative hypothesis. This suggests that quitting smoking leads to a positive change in weight.