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State the fictional assumptions
| Input | Value |
|---|---|
| Sample mean | 50.4 |
| Null mean | 50 |
| Known population standard deviation | 2 |
| Independent normal observations | 100 |
| Planned alternative | Two-sided |
| Planned significance level | 5% |
Calculate uncertainty before standardizing
The standard error is 2/√100 = 0.2 measurement units. The estimated difference is 50.4 − 50 = 0.4 units. Dividing gives z = 0.4/0.2 = 2. The numerator and denominator use the same units, so z itself is dimensionless.
Read the two-sided tail area
For z = 2, the upper standard-normal tail is approximately 0.0227501. The symmetric two-sided p-value is approximately 0.0455003. With the stated p ≤ alpha convention, this is below the planned 0.05 threshold and the exercise rejects the null.
Keep the conclusion tied to the model
The calculation concerns the assumed mean and sampling model. It has not shown that the difference of 0.4 units is economically important, that a treatment caused it, or that the observations are actually independent. If 2 were an estimated sample standard deviation, the exact normal reference used here would no longer follow from these inputs.