Answer :

When conducting a hypothesis test using a t-test, you are comparing a calculated t value to a critical t value to determine if you should reject the null hypothesis.

Here's a step-by-step explanation:

  1. Understanding t-test: A t-test is used to determine if there is a significant difference between the means of two groups. It involves calculating a t statistic from the data and comparing it to a critical value from the t distribution, based on your significance level and degrees of freedom.

  2. Given Values:

    • Calculated t value: [tex]t = 1.69[/tex]
    • Critical t value: [tex]t_{critical} = 2.02[/tex]

  3. Decision Rule:

    • If the absolute value of the calculated t value [tex]|t|[/tex] is greater than the critical t value [tex]t_{critical}[/tex], you reject the null hypothesis. In simpler terms, [tex]|t| > t_{critical}[/tex] means there is statistically significant evidence to support the alternative hypothesis.
    • If [tex]|t|[/tex] is less than or equal to [tex]t_{critical}[/tex], you fail to reject the null hypothesis, meaning there is no significant difference based on the data.

  4. Applying the Decision Rule:

    • Here, [tex]|1.69|[/tex] is less than [tex]2.02[/tex].
    • Therefore, you fail to reject the null hypothesis.
    • This implies that, based on the data and the significance level you are using, there is not enough evidence to conclude that there is a significant difference between the means of the two groups being studied.

  5. Next Steps:

    • Consider the context of the study to determine any other potential factors that may affect the outcome.
    • If necessary, reevaluate the sample size or consider additional data to strengthen the analysis.

This understanding helps you interpret the results of hypothesis testing and make decisions based on statistical analysis.

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