Statistical Inference and Hypothesis Testing
2026-02-21
Statistical inference is the process of drawing conclusions about a population based on data from a sample.
| Term | Symbol | Example |
|---|---|---|
| Population mean | \(\mu\) | Avg height of all students in PH |
| Sample mean | \(\bar{x}\) | Avg height of 50 students surveyed |
| Population proportion | \(p\) | % of voters who prefer candidate A |
| Sample proportion | \(\hat{p}\) | % in a poll of 200 voters |
A sampling distribution is the distribution of a statistic (like \(\bar{x}\)) computed from many samples of the same size.
A sampling distribution is the distribution of a statistic (like \(\bar{x}\)) computed from many samples of the same size.
Special case of sampling distribution:
Special case of sampling distribution:
The Central Limit Theorem (CLT)
For large enough \(n\), the sampling distribution of \(\bar{x}\) is approximately normal, regardless of the population shape.
\[\bar{x} \sim N\!\left(\mu,\; \frac{\sigma}{\sqrt{n}}\right)\]
Special note about CLT
As sample size increases:
Worked Example
A population has mean \(\mu = 70\) and \(\sigma = 10\). We take samples of size \(n = 25\). Find the mean and SE of the sampling distribution.
\[\text{Mean} = \mu = 70\]
\[SE = \frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{25}} = \frac{10}{5} = 2\]
So \(\bar{x} \sim N(70,\; 2)\). Most sample means will fall between 66 and 74.
A confidence interval (CI) gives a range of plausible values for a population parameter.
\[\bar{x} \pm z^* \cdot \frac{\sigma}{\sqrt{n}}\]
Common CI levels:
Common Misconception:
Example:
Example:
A sample of \(n = 36\) students has \(\bar{x} = 68\) kg and \(\sigma = 12\) kg. Construct a 95% CI for the population mean.
\[\bar{x} \pm z^* \cdot \frac{\sigma}{\sqrt{n}} = 68 \pm 1.96 \cdot \frac{12}{\sqrt{36}} = 68 \pm 1.96 \cdot 2 = 68 \pm 3.92\]
\[\boxed{(64.08,\; 71.92)}\]
We are 95% confident the true mean weight is between 64.08 kg and 71.92 kg.
Hypothesis testing is a formal procedure to decide whether sample evidence is strong enough to reject a claim about a population.
Hypothesis testing is a formal procedure to decide whether sample evidence is strong enough to reject a claim about a population.
Two Hypotheses
Step-by-Step Process
Step-by-Step Process
Tip
\(H_0\) always contains =. \(H_a\) usually opposes \(H_0\), and usually uses \(\neq\), \(<\), or \(>\).
“Is the mean different from 50?”
\[H_0: \mu = 50\] \[H_a: \mu \neq 50\]
“Is the mean less than 50?”
\[H_0: \mu = 50\] \[H_a: \mu < 50\]
“Is the mean greater than 50?”
\[H_0: \mu = 50\] \[H_a: \mu > 50\]
Note
The direction of \(H_a\) determines where the rejection region falls in the distribution.
The p-value is the probability of getting a result as extreme as your sample, assuming \(H_0\) is true.
The p-value is the probability of getting a result as extreme as your sample, assuming \(H_0\) is true.
Common significance levels: \(\alpha = 0.05\) or \(\alpha = 0.01\)
| p-value | Evidence against \(H_0\) |
|---|---|
| \(> 0.10\) | Weak |
| \(0.05 - 0.10\) | Moderate |
| \(0.01 - 0.05\) | Strong |
| \(< 0.01\) | Very strong |
Important
Failing to reject \(H_0\) does not prove \(H_0\) is true – it just means there is not enough evidence to reject it.
When we make a decision, we can be wrong in two ways:
| \(H_0\) is True | \(H_0\) is False | |
|---|---|---|
| Reject \(H_0\) | Type I Error (\(\alpha\)) | Correct (Power) |
| Fail to Reject \(H_0\) | Correct | Type II Error (\(\beta\)) |
Remarks:
Use the Z-test when: population \(\sigma\) is known and \(n \geq 30\) (or population is normal).
Use the Z-test when: population \(\sigma\) is known and \(n \geq 30\) (or population is normal).
\[z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}\]
Use the Z-test when: population \(\sigma\) is known and \(n \geq 30\) (or population is normal).
\[z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}\]
Worked Example
A school claims the average test score is \(\mu = 75\). A sample of \(n = 40\) gives \(\bar{x} = 78\), with known \(\sigma = 10\). Test at \(\alpha = 0.05\) (two-tailed).
\[z = \frac{78 - 75}{10/\sqrt{40}} = \frac{3}{1.581} \approx 1.897\]
Critical value: \(z^* = \pm 1.96\). Since \(|1.897| < 1.96\), we fail to reject \(H_0\). There is insufficient evidence that the mean differs from 75.
Use the T-test when: population \(\sigma\) is unknown (use sample \(s\) instead) or \(n < 30\).
\[t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}, \quad df = n - 1\]
Use the T-test when: population \(\sigma\) is unknown (use sample \(s\) instead) or \(n < 30\).
\[t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}, \quad df = n - 1\]
Worked Example
A sample of \(n = 16\) bags shows \(\bar{x} = 498\) g and \(s = 6\) g. Claim: \(\mu = 500\) g. Test at \(\alpha = 0.05\) (left-tailed).
\[t = \frac{498 - 500}{6/\sqrt{16}} = \frac{-2}{1.5} \approx -1.333\]
Critical value (\(df = 15\), left-tailed): \(t^* = -1.753\). Since \(-1.333 > -1.753\), we fail to reject \(H_0\). No significant evidence that bags are underfilled.
flowchart TD
A[Start: Testing a mean] --> B{Is sigma known?}
B -->|Yes| C{n >= 30?}
B -->|No| E{n >= 30?}
C -->|Yes| F[Z-test]
C -->|No| G{Population normal?}
G -->|Yes| F
G -->|No| H[Need larger sample]
E -->|Yes| I[T-test, approx. normal]
E -->|No| J{Population normal?}
J -->|Yes| K[T-test, df = n-1]
J -->|No| H
Use a proportion test when the variable is categorical (yes/no, success/failure).
\[z = \frac{\hat{p} - p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}\]
Conditions: \(np_0 \geq 10\) and \(n(1-p_0) \geq 10\)
Use a proportion test when the variable is categorical (yes/no, success/failure).
\[z = \frac{\hat{p} - p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}\]
Conditions: \(np_0 \geq 10\) and \(n(1-p_0) \geq 10\)
Worked Example
A candidate claims 60% support (\(p_0 = 0.60\)). A poll of \(n = 200\) finds \(\hat{p} = 0.55\). Test at \(\alpha = 0.05\) (left-tailed).
\[z = \frac{0.55 - 0.60}{\sqrt{0.60 \cdot 0.40 / 200}} = \frac{-0.05}{0.03464} \approx -1.443\]
Critical value: \(z^* = -1.645\). Since \(-1.443 > -1.645\), we fail to reject \(H_0\). Not enough evidence that support is below 60%.
Before running a proportion test, verify:
| Condition | Why? |
|---|---|
| Random sample | Avoids bias |
| Independence: \(n \leq 10\%\) of population | Observations are independent |
| Success: \(np_0 \geq 10\) | Normal approximation is valid |
| Failure: \(n(1-p_0) \geq 10\) | Normal approximation is valid |
Tip
If conditions are not met, the z-approximation may be unreliable – results should be interpreted with caution.