In the cookie-calorie one-sample $t$-test example, what is the two-tailed critical value for $9$ degrees of freedom at the .05 significance level?

In the cookie-calorie study, switching from a two-tailed to a one-tailed test makes it easier to reject the null hypothesis when results trend in the predicted direction, because the one-tailed critical value ($-1.833$) is less extreme than the two-tailed critical value ($\pm 2.262$).

Match each element from the cookie-calorie one-sample $t$-test study to the role it plays in the statistical decision-making process.

In the health psychologist's cookie-calorie study, because the sample mean estimate of $212.00$ is lower than the null hypothesis value of $250$ and the calculated $t$ score is more extreme than the critical value, the researcher rejects the null hypothesis and concludes that students significantly _____ the cookie's calories.

Order the following steps in evaluating the results of the cookie-calorie one-sample $t$-test, from formulating the initial hypothesis to making the final scientific decision.

In the health psychologist's cookie-calorie example, what was the specific null hypothesis ($\mu$) value representing accurate estimation?

In the context of the cookie-calorie estimation study, explain the role of the null hypothesis ($\mu = 250$) and describe how the researcher uses the calculated $t$ score of $-3.07$ and the critical value of $\pm 2.262$ to make a statistical decision about student estimation accuracy.

Based on the case context, if the researcher changes the analysis to a one-tailed test with a critical value of $-1.833$, how does this modify the critical value threshold, how does it affect the ease of rejecting the null hypothesis, and what is the final conclusion regarding the students' estimations?

Analyze the relationship between the sample size ($N = 10$) and the degrees of freedom ($df = 9$) in this one-sample $t$-test. How would an increase in sample size affect the degrees of freedom and the critical value needed to reject the null hypothesis?