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hypothesis testing with critical values

Instead of using a p‑value, you compare your test statistic (like a z‑score or t‑score) to a critical value that marks the boundary of the rejection region.

If your test statistic falls beyond the critical value → reject H_0.
If it falls inside the non‑rejection region → fail to reject H_0.

It’s like checking whether your result is “extreme enough” to count as evidence.

The Steps

1. State the hypotheses

Example:
H_0: \mu = 50 \quad\text{vs.}\quad H_a: \mu \neq 50

2. Choose a significance level \alpha

Common: 0.05, 0.01, 0.10.

3. Find the critical value(s)

Depends on:

  • the test (z or t)
  • whether it’s one‑tailed or two‑tailed
  • the chosen \alpha

4. Compute the test statistic

Example for a t‑test:
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}

5. Compare test statistic to critical value

  • If |t| > t_{\text{crit}} → reject H_0
  • Otherwise → fail to reject H_0

⭐ Example 1: Two‑Tailed t‑Test (Mean Difference)

A company claims the average weight of their cereal boxes is 500 g.

You sample 16 boxes:

  • \bar{x} = 492
  • s = 20

Test:
H_0: \mu = 500 \quad\text{vs.}\quad H_a: \mu \neq 500

Step 1: Choose \alpha = 0.05

Step 2: Critical value

Two‑tailed, df = 15 →
t_{\text{crit}} = \pm 2.131

Step 3: Compute test statistic

t = \frac{492 - 500}{20/\sqrt{16}} = \frac{-8}{5} = -1.6

Step 4: Compare

-1.6 \text{ is NOT beyond } -2.131

Decision: Fail to reject H_0.

No significant evidence that the mean differs from 500 g.

⭐ Example 2: One‑Tailed z‑Test (Proportion)

A website claims 40% of visitors click the “Subscribe” button.

You sample 200 visitors and find 38% clicked.

Test:
H_0: p = 0.40 \quad\text{vs.}\quad H_a: p < 0.40

Step 1: \alpha = 0.05

Step 2: Critical value

One‑tailed →
z_{\text{crit}} = -1.645

Step 3: Compute test statistic

z = \frac{0.38 - 0.40}{\sqrt{0.40(0.60)/200}} \approx -0.91

Step 4: Compare

-0.91 \text{ is NOT less than } -1.645

Decision: Fail to reject H_0.

Not enough evidence that the true click rate is lower than 40%.

⭐ Example 3: One‑Tailed t‑Test (Before/After)

A teacher believes a new study technique increases test scores.

Differences (after − before) for 10 students give:

  • \bar{d} = 4.5
  • s_d = 6.0
See also  Type I and type II errors

Test:
H_0: \mu_d \le 0 \quad\text{vs.}\quad H_a: \mu_d > 0

Step 1: \alpha = 0.05

Step 2: Critical value

One‑tailed, df = 9 →
t_{\text{crit}} = 1.833

Step 3: Compute test statistic

t = \frac{4.5}{6/\sqrt{10}} = \frac{4.5}{1.897} \approx 2.37

Step 4: Compare

2.37 > 1.833

Decision: Reject H_0.

Evidence suggests the study technique improves scores.

Why students like the critical‑value method

It feels visual and rule‑based:

  • Draw the distribution
  • Mark the rejection region
  • Drop the test statistic in
  • See where it lands

It’s like checking whether your result crosses a finish line.

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