Why Alpha and Beta Inverse in Hypothesis errors

Disclaimer: Gemini Response

To understand why $\alpha$ and $\beta$ are inversely related and how $\beta$ is calculated, it helps to picture two overlapping bell curves:

1. The Null Distribution (assuming $H_0$ is true).
2. The Alternative Distribution (assuming a specific true effect size exists, $H_A$).

1. Why $\alpha$ and $\beta$ are Inversely Related

$\alpha$ and $\beta$ are inversely related because they are separated by a single decision threshold (the critical value line) on your measurement scale.

• $\alpha$ (Type I Error) is the area under the Null Curve that falls past the critical line into the "Reject $H_0$" region.
• $\beta$ (Type II Error) is the area under the Alternative Curve that falls behind the critical line into the "Fail to Reject $H_0$" region.

               [ Null Distribution (H0) ]            [ Alternative Distribution (HA) ]
                          |                                     |
                          |        Critical Threshold           |
                          |                 |                   |
                        ____                |                 ____
                       /    \               |                /    \
                      /      \              |               /      \
                     /        \  Alpha (α)  |   Beta (β)   /        \
                    /          \   [--->]   |   [<---]    /          \
                  --------------------------+--------------------------
                                            |
                                      Move line Left:  α ↑ , β ↓
                                      Move line Right: α ↓ , β ↑

The Mechanism

• If you want to strictly avoid Type I errors, you make it harder to reject $H_0$. You lower $\alpha$ (e.g., from $0.05$ to $0.01$). This shifts the critical line to the right.
• By shifting the line right, you shrink the $\alpha$ tail on the Null distribution—but you expand the area captured under the Alternative curve on the non-rejection side. Thus, $\beta$ increases.
• Conversely, if you make it easier to reject $H_0$ by increasing $\alpha$, you move the line to the left, which shrinks the non-rejection area under the Alternative curve, so $\beta$ decreases.

Note: They do not necessarily sum to $1$ (e.g., $\alpha = 0.05$ does not mean $\beta = 0.95$). Their exact mathematical balance depends on sample size ($n$), population variance ($\sigma$), and the true effect size.

2. How $\beta$ is Calculated

Calculating $\beta$ requires knowing or assuming a specific true alternative value ($\mu_A$) for the population parameter, as well as the sample size ($n$), population standard deviation ($\sigma$), and significance level ($\alpha$).

It is computed in a 4-step process:

Step 1: Define the Hypotheses and Parameters

Assume a right-tailed Z-test:

• $H_0$: $\mu = \mu_0$ (e.g., $100$)

• $H_A$: $\mu = \mu_A$ (e.g., a specific true mean, like $105$)

• Standard Error ($SE$) = $\frac{\sigma}{\sqrt{n}}$

Step 2: Find the Critical Value ($\bar{x}_{\text{crit}}$) under $H_0$

Find the cutoff sample mean that corresponds to your chosen $\alpha$ on the null distribution:

(Any sample mean $\bar{x} > \bar{x}_{\text{crit}}$ leads to rejecting $H_0$.)

Step 3: Standardize $\bar{x}_{\text{crit}}$ relative to the True Alternative Mean ($\mu_A$)

Calculate how many standard errors $\bar{x}_{\text{crit}}$ is away from the actual true mean $\mu_A$:

Step 4: Calculate the Area ($\beta$)

$\beta$ is the probability of failing to reject $H_0$ when $\mu = \mu_A$. For a right-tailed test, this is the area to the left of $Z_{\text{alt}}$ on the standard normal distribution:

Concrete Example

Suppose:

• $H_0: \mu = 100$, $H_A: \mu = 105$

• $\sigma = 10$, $n = 25 \implies SE = \frac{10}{\sqrt{25}} = 2$

• Set $\alpha = 0.05 \implies Z_{0.05} = 1.645$

1. Find $\bar{x}_{\text{crit}}$:

2. Find $Z_{\text{alt}}$ relative to $\mu_A = 105$:

3. Calculate $\beta$ using standard normal table:

Result: With $\alpha = 0.05$, there is a 19.63% chance ($\beta$) of making a Type II error if the true population mean is actually $105$. The statistical power ($1 - \beta$) would be 80.37%.