Test if one value comes from N(μ,σ²)

The Problem

Suppose you have a production process that is monitored by a sensor. Based on historical data, the sensor readings follow a normal distribution $N(\mu, \sigma^2)$ where $\mu = 100$ and $\sigma = 10$ are known. A single new reading $x = 125$ is observed.

1. Formulate a null hypothesis $H_0$ and an alternative hypothesis $H_a$ to test whether this specific observation is consistent with the known distribution.
2. Calculate the test statistic (z-score) and the corresponding p-value for a two-sided test.
3. Using a significance level of $\alpha = 0.05$, state your decision rule and conclude whether you reject $H_0$.

Constraints and Considerations

• Explain the assumptions required to perform this test on a sample size of $n=1$.
• Discuss why the power of this test is inherently limited given the sample size.
• How would the decision rule change if the business required a stricter control over false alarms (e.g., $\alpha = 0.01$)?

Example

If $\mu=50, \sigma=5$ and $x=62$, then:

• $z = (62 - 50) / 5 = 2.4$.
• The two-sided p-value is $2 * P(Z > |2.4|) \approx 0.0164$.
• At $\alpha = 0.05$, we reject $H_0$ because $0.0164 < 0.05$.