Day 24

Math 216: Statistical Thinking

Bastola

Steps for Hypothesis Testing Part I:

State the Hypotheses
- Null Hypothesis ($H_0$): No effect or no difference, posits that $\mu = \mu_0$.
- Alternative Hypothesis ($H_A$): Asserts a specific claim about the population mean $\mu$ that is different from $\mu_0$ (e.g., $\mu > \mu_0$, $\mu < \mu_0$, or $\mu \neq \mu_0$).
Set the Significance Level ($\alpha$)
- The probability of making a Type I error (rejecting a true null hypothesis). Common values are 0.05, 0.01, or 0.10.
Assumptions
- The sample is a simple random sample.
- The sample size is large (usually $n \geq 30$), which justifies the use of the Central Limit Theorem.
Collect Data
- Obtain a random sample and compute the sample mean ($\bar{X}$) and standard deviation ($S$).

Calculate the Test Statistic ($z$)
- If the population standard deviation ($\sigma$) is known: \[ z = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}} \]
- If $\sigma$ is unknown and $n \geq 30$, use $s$ (sample standard deviation): \[ z = \frac{\bar{X} - \mu_0}{s/\sqrt{n}} \]
Decision Rule
- For a one-tailed test: Reject $H_0$ if $z$ falls in the rejection region (either $z < -z_{\alpha}$ for left-tailed or $z > z_{\alpha}$ for right-tailed).
- For a two-tailed test: Reject $H_0$ if $|z| > z_{\alpha/2}$.
Compute the P-Value
- The probability of observing a test statistic as extreme as, or more extreme than, the statistic calculated from the sample data, assuming that $H_0$ is true.
Conclusion
- Reject $H_0$ if the p-value is less than $\alpha$; otherwise, do not reject $H_0$. Report whether the evidence supports $H_a$ based on the statistical significance.

Scenario: A study group claims that their session increases the average test score above 75.
Sample Data: $\bar{x} = 78$, $s = 5$, $n = 30$
Test: $H_0: \mu \leq 75$ vs. $H_a: \mu > 75$
Calculation: $z = \frac{78 - 75}{5/\sqrt{30}} \approx 3.29$
Conclusion: Reject $H_0$ if $z > 1.645$ (at $\alpha = 0.05$).

Scenario: A nutritionist claims that a new diet reduces the average calorie intake below 2200 calories/day.
Sample Data: $\bar{x} = 2150$, $s = 120$, $n = 35$
Test: $H_0: \mu \geq 2200$ vs. $H_a: \mu < 2200$
Calculation: $z = \frac{2150 - 2200}{120/\sqrt{35}} \approx -2.47$
Conclusion: Reject $H_0$ if $z < -1.645$ (at $\alpha = 0.05$).

Scenario: An analyst claims that a new billing system changes the average monthly bill from $100.
Sample Data: $\bar{x} = 105$, $s = 10$, $n = 40$
Test: $H_0: \mu = 100$ vs. $H_a: \mu \neq 100$
Calculation: $z = \frac{105 - 100}{10/\sqrt{40}} \approx 3.16$
Conclusion: Reject $H_0$ if $|z| > 1.96$ (at $\alpha = 0.05$).