7-Step Hypothesis Testing Procedure
Use this exact procedure for every hypothesis test in MTH 160.
-
State the original claim and identify \(H_0\) and \(H_1\) The null hypothesis \(H_0\) always contains equality (\(=\), \(\le\), or \(\ge\)). The alternative \(H_1\) is what you're testing for (\(\ne\), \(<\), or \(>\)).
-
State the significance level \(\alpha\) Common values: \(\alpha = 0.05\) (5%) or \(\alpha = 0.01\) (1%). This is the probability of a Type I error you are willing to accept.
-
Identify the test statistic and its sampling distribution Use \(z\) when \(\sigma\) is known or \(n \ge 30\). Use \(t\) when \(\sigma\) is unknown and \(n < 30\). For proportions, use \(z\).
-
Find the critical value(s) and state the rejection region For a two-tailed test split \(\alpha/2\). Look up the critical value in the z-table or t-table.
-
Compute the test statistic Use the appropriate formula based on the type of test (mean, proportion, etc.).
-
Make the statistical decision If the test statistic falls in the rejection region (or p-value < \(\alpha\)), reject \(H_0\). Otherwise, fail to reject \(H_0\).
-
State the conclusion in plain English Always reference the original claim. Say "There is (not) sufficient evidence to support the claim that…"
Tail Type Reference
| Test Type | H₁ symbol | Rejection Region |
|---|---|---|
| Left-tailed | \(H_1: \mu < k\) | Left tail: reject if \(z < -z_\alpha\) |
| Right-tailed | \(H_1: \mu > k\) | Right tail: reject if \(z > z_\alpha\) |
| Two-tailed | \(H_1: \mu \ne k\) | Both tails: reject if \(|z| > z_{\alpha/2}\) |
MTH 160 Formula Reference
Descriptive Statistics
Sample Mean
\[\bar{x} = \frac{\sum x}{n}\]Sample Variance
\[s^2 = \frac{\sum(x-\bar{x})^2}{n-1}\]Sample Std Dev
\[s = \sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}\]Population Mean
\[\mu = \frac{\sum x}{N}\]Population Std Dev
\[\sigma = \sqrt{\frac{\sum(x-\mu)^2}{N}}\]Z-Score
\[z = \frac{x - \mu}{\sigma}\]Hypothesis Testing — Means
z-test (σ known)
\[z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}\]t-test (σ unknown)
\[t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}, \; df = n-1\]Standard Error
\[SE = \frac{\sigma}{\sqrt{n}}\]Hypothesis Testing — Proportions
Sample Proportion
\[\hat{p} = \frac{x}{n}\]z-test for Proportion
\[z = \frac{\hat{p} - p_0}{\sqrt{p_0 q_0 / n}}\]where \(q_0 = 1 - p_0\)
Confidence Intervals
CI for Mean (σ known)
\[\bar{x} \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\]CI for Mean (σ unknown)
\[\bar{x} \pm t_{\alpha/2}\frac{s}{\sqrt{n}}\]CI for Proportion
\[\hat{p} \pm z_{\alpha/2}\sqrt{\frac{\hat{p}\hat{q}}{n}}\]Margin of Error (mean)
\[E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\]Sample Size (mean)
\[n = \left(\frac{z_{\alpha/2}\cdot\sigma}{E}\right)^2\]Sample Size (proportion)
\[n = \hat{p}\hat{q}\left(\frac{z_{\alpha/2}}{E}\right)^2\]Binomial Distribution
Binomial Probability
\[P(x) = \binom{n}{x}p^x q^{n-x}\]Binomial Mean
\[\mu = np\]Binomial Std Dev
\[\sigma = \sqrt{npq}\]Combination Formula
\[\binom{n}{x} = \frac{n!}{x!(n-x)!}\]Correlation & Regression
Pearson r
\[r = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2-(\sum x)^2][n\sum y^2-(\sum y)^2]}}\]Regression Line
\[\hat{y} = b_0 + b_1 x\]Slope
\[b_1 = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}\]Intercept
\[b_0 = \bar{y} - b_1\bar{x}\]Common Critical Values
90% CI: z = 1.645
95% CI: z = 1.96
98% CI: z = 2.33
99% CI: z = 2.575
Type I & Type II Errors
| Decision | H₀ is Actually True | H₀ is Actually False |
|---|---|---|
| Reject H₀ | Type I Error (α) False Positive "Convicting an innocent person" | Correct Decision Power = 1 − β |
| Fail to Reject H₀ | Correct Decision Confidence = 1 − α | Type II Error (β) False Negative "Letting a guilty person go free" |
Key Relationships
- α (alpha) = P(Type I Error) = significance level. Set by the researcher before the test.
- β (beta) = P(Type II Error). Decreases as sample size increases.
- Power = 1 − β = probability of correctly rejecting a false H₀.
- Decreasing α increases β (the two errors trade off).
- Increasing sample size \(n\) decreases both α and β simultaneously.
Example Scenarios
| Scenario | Type I Error Meaning | Type II Error Meaning |
|---|---|---|
| Drug testing | Approve an ineffective drug | Reject an effective drug |
| Quality control | Reject a good batch | Accept a defective batch |
| Criminal trial | Convict an innocent person | Acquit a guilty person |
Interactive Z-Score Calculator
Result:
Interpreting z-scores
- \(|z| < 1\): within 1 standard deviation of the mean (common)
- \(1 \le |z| < 2\): somewhat unusual
- \(|z| \ge 2\): unusual (outside 95% of normal data)
- \(|z| \ge 3\): very unusual (outside 99.7% of normal data)
Standard Normal Z-Table
Values represent the cumulative area to the left of z. The table covers z = −3.4 to +3.4.
Negative Z-Values (z < 0)
| z | .00 | .01 | .02 | .03 | .04 | .05 | .06 | .07 | .08 | .09 |
|---|
Positive Z-Values (z ≥ 0)
| z | .00 | .01 | .02 | .03 | .04 | .05 | .06 | .07 | .08 | .09 |
|---|
t-Distribution Critical Values (Table A-3)
Values are critical t-values for the given degrees of freedom (df) and significance level α (two-tailed).
| df | Two-Tailed α | One-Tailed α | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 0.20 | 0.10 | 0.05 | 0.02 | 0.01 | 0.10 | 0.05 | 0.025 | 0.01 | 0.005 | |
For df > 100 use z critical values: z₀.₀₅ = 1.645, z₀.₀₂₅ = 1.96, z₀.₀₁ = 2.33, z₀.₀₀₅ = 2.575
Pearson r Critical Values (Table A-6)
If |r| exceeds the critical value, there is significant linear correlation (α = 0.05, two-tailed).
| n | α = 0.05 | α = 0.01 |
|---|
Binomial Probability Table (Table A-1)
Values show \(P(X = x)\) for given \(n\) and \(p\). Select \(n\) to display its table.
Select n and choose Update Binomial Table to display the probability table.
Formula Reminder
\[P(X = x) = \binom{n}{x}p^x(1-p)^{n-x}\]To find \(P(X \le k)\): sum the probabilities for \(x = 0, 1, \ldots, k\).
🎯 Statistics Quiz
8 questions — click your answer to check instantly.
1. In hypothesis testing, the null hypothesis H₀ always contains:
2. A Type I error occurs when:
3. A data value has \(x = 85\), \(\mu = 75\), \(\sigma = 10\). Its z-score is:
4. For a 95% confidence interval for a mean, the critical z-value is:
5. When σ is unknown and n < 30, the correct test statistic for a mean is:
6. In the binomial distribution with n = 10 and p = 0.3, the mean is:
7. The power of a test is:
8. A two-tailed test at α = 0.05 rejects H₀ when: