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Solution Methods — Inference Using Normal and t-Distributions

Method 1: One-Sample t-Test Procedure​

适用于：单个总体均值检验，方差未知，样本量小。

Steps

Step 1: State hypotheses H_0: \mu = \mu_0, \quad H_1: \mu \neq \mu_0 \ (\text{or } > \ \text{or } <)

Step 2: Calculate test statistic t=xˉ−μ0s2/nt = \frac{\bar{x} - \mu_0}{\sqrt{s^2/n}}

Step 3: Find critical value From t-tables: tα/2(n−1)t_{\alpha/2}(n-1) (two-tail) or tα(n−1)t_{\alpha}(n-1) (one-tail)

Step 4: Compare and conclude If |t| > t_{\text{crit}}, reject H0H_0; otherwise do not reject H0H_0.

Method 2: Two-Sample Pooled t-Test Procedure​

适用于：两个独立总体均值比较，方差相等但未知，样本量小。

Steps

Step 1: State hypotheses H_0: \mu_1 = \mu_2, \quad H_1: \mu_1 \neq \mu_2 \ (\text{or } > \ \text{or } <)

Step 2: Pool variances sp2=(n1−1)s12+(n2−1)s22n1+n2−2s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}

Step 3: Calculate test statistic t=xˉ1−xˉ2sp2(1/n1+1/n2)t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_p^2(1/n_1 + 1/n_2)}}

Step 4: Find critical value From t-tables: tα/2(n1+n2−2)t_{\alpha/2}(n_1+n_2-2) or tα(n1+n2−2)t_{\alpha}(n_1+n_2-2)

Step 5: Compare and conclude

Method 3: Paired t-Test Procedure​

适用于：同一组对象前后两次测量 / 配对设计。

Steps

Step 1: Compute differences di=xi−yi for each paird_i = x_i - y_i \ \text{for each pair}

Step 2: Calculate mean and variance of differences dˉ=∑din,sd2=∑(di−dˉ)2n−1\bar{d} = \frac{\sum d_i}{n}, \quad s_d^2 = \frac{\sum(d_i - \bar{d})^2}{n-1}

Step 3: State hypotheses H_0: \mu_d = d_0, \quad H_1: \mu_d \neq d_0 \ (\text{or } > \ \text{or } <)

Step 4: Calculate test statistic t=dˉ−d0sd2/nt = \frac{\bar{d} - d_0}{\sqrt{s_d^2/n}}

Step 5: Find critical value From t-tables: tα/2(n−1)t_{\alpha/2}(n-1) or tα(n−1)t_{\alpha}(n-1)

Step 6: Compare and conclude

Method 4: Large-Sample z-Test Procedure​

适用于：大样本（n≥30n \ge 30）情况下的均值检验。

Steps

Step 1: State hypotheses

Step 2: Calculate z-statistic One-sample: z=xˉ−μ0σ/nz = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} or z=xˉ−μ0s/nz = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} Two-sample: z=xˉ1−xˉ2s12/n1+s22/n2z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_1^2/n_1 + s_2^2/n_2}}

Step 3: Compare with standard normal critical values z0.05=1.645z_{0.05} = 1.645, z0.025=1.960z_{0.025} = 1.960, z0.005=2.576z_{0.005} = 2.576

Step 4: Conclude

Method 5: Confidence Interval Construction​

Steps

One-sample (t): xˉ±tα/2(n−1)⋅s2/n\bar{x} \pm t_{\alpha/2}(n-1) \cdot \sqrt{s^2/n}

Two-sample pooled (t): (xˉ1−xˉ2)±tα/2(n1+n2−2)⋅sp2(1/n1+1/n2)(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2}(n_1+n_2-2) \cdot \sqrt{s_p^2(1/n_1 + 1/n_2)}

Paired (t): dˉ±tα/2(n−1)⋅sd2/n\bar{d} \pm t_{\alpha/2}(n-1) \cdot \sqrt{s_d^2/n}

Two-sample large (z): (xˉ1−xˉ2)±zα/2⋅s12/n1+s22/n2(\bar{x}_1 - \bar{x}_2) \pm z_{\alpha/2} \cdot \sqrt{s_1^2/n_1 + s_2^2/n_2}

Quick Decision Tree​

Is it one sample or two?
├── One sample
│   ├── Small n (n < 30), σ² unknown → One-sample t-test (Method 1)
│   └── Large n (n ≥ 30) → z-test (Method 4)
└── Two samples
    ├── Paired (same subjects) → Paired t-test (Method 3)
    └── Independent
        ├── Small n, equal σ² → Pooled t-test (Method 2)
        └── Large n → z-test (Method 4)