Which of the following is the correct formula for calculating the $t$ statistic in a one-sample $t$-test?

In the one-sample $t$-test formula ($t = \frac{M - \mu_0}{\frac{SD}{\sqrt{N}}}$), each mathematical component represents a specific concept used in psychological research. Match each term with the concept it quantifies.

Based on the mathematical relationships in the one-sample $t$-test formula $t = \frac{M - \mu_0}{\frac{SD}{\sqrt{N}}}$, arrange the following research scenarios from the one that results in the smallest absolute $t$ value (order 1) to the one that results in the largest absolute $t$ value (order 4).

A researcher evaluates two potential modifications to a study design to increase the magnitude of the $t$ statistic ($t = \frac{M - \mu_0}{\frac{SD}{\sqrt{N}}}$). Modification A is to quadruple the sample size ($N$), and Modification B is to double the difference between the sample mean and the population mean ($M - \mu_0$). The researcher concludes that both modifications will have an identical impact on the calculated $t$ value, specifically doubling its magnitude. This evaluation is correct.

In the formula for a one-sample $t$-test, the sample mean ($M$) is subtracted from the hypothetical population mean ($\mu_0$) to calculate the numerator.

In the one-sample $t$-test formula $t = \frac{M - \mu_0}{\frac{SD}{\sqrt{N}}}$, what does the final calculated $t$ value represent conceptually?

A cognitive psychologist is testing a new memory-enhancement technique. The average number of words recalled by the general population on a specific test is 15. A sample of 36 students using the new technique recalls an average of 18 words with a sample standard deviation of 9. Using the one-sample $t$-test formula $t = \frac{M - \mu_0}{\frac{SD}{\sqrt{N}}}$, the $t$ statistic for this sample is _____.

A clinical psychology researcher wants to test whether an intervention reduces anxiety scores below the national average. The hypothetical population mean anxiety score is $\mu_0 = 50$. After the intervention, a sample of $N = 25$ participants has a mean anxiety score of $M = 44$ with a sample standard deviation of $SD = 10$. Using the one-sample $t$-test formula $t = \frac{M - \mu_0}{\frac{SD}{\sqrt{N}}}$, calculate the $t$ statistic. Show your work step-by-step and explain how the numerator and the denominator each contribute to the final calculated value.

Analyze Dr. Smith's error in the calculation. How does failing to take the square root of the sample size ($N$) mathematically distort the denominator of the $t$ statistic formula, and what impact does this specific error have on the resulting $t$ value? Provide the correct calculation to support your analysis.

A researcher is evaluating two strategies to try and increase the absolute value of their $t$ statistic in a one-sample $t$-test. Strategy A involves refining their measurement tool to decrease the sample standard deviation ($SD$), while Strategy B involves purposefully reducing the sample size ($N$) to save money. Evaluate which strategy is mathematically sound for increasing the absolute value of $t$ based on the formula $t = \frac{M - \mu_0}{\frac{SD}{\sqrt{N}}}$, and briefly explain why.

Using the one-sample $t$-test formula $t = \frac{M - \mu_0}{\frac{SD}{\sqrt{N}}}$, calculate the $t$ statistic for each of the following hypothetical psychological research studies, and match each study with its correct calculated $t$ value.

A cognitive psychologist analyzes the mathematical relationships within the one-sample $t$-test formula: $t = \frac{M - \mu_0}{\frac{SD}{\sqrt{N}}}$. If the psychologist quadruples the sample size ($N$) while keeping the sample mean ($M$), the hypothetical population mean ($\mu_0$), and the sample standard deviation ($SD$) completely unchanged, the final calculated $t$ statistic will be multiplied by ____.

An institutional review board asks a research team to evaluate three different internal critiques regarding their statistical plan for a one-sample $t$-test ($t = \frac{M - \mu_0}{\frac{SD}{\sqrt{N}}}$).

Arrange these critiques in order from the most scientifically and mathematically valid (Order 1) to the least valid/most flawed (Order 3).