Hypothesis Testing For A Population Mean Sample

Introduction & Statement of Hypotheses

This statistical analysis evaluates a local factory's claim regarding the average weight of cereal boxes. The manufacturer states a population mean weight of 500 grams. Given suspected variations in the automated filling process implemented in early 2024, a formal hypothesis test for a single population mean determines if the actual average is significantly lower. The null hypothesis (H₀) represents the factory's claim, while the alternative hypothesis (H_a) models a systematic under-filling scenario. To evaluate the results, a standard industrial alpha level (α) of 0.05 is applied (Moore et al., 2021).

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H₀: μ = 500 grams
H_a: μ < 500 grams

Data Summary & Assumption Checks

A random sample of n = 30 cereal boxes collected on October 15, 2024, yielded a sample mean (x̄) of 495 grams. Historical calibration data from the manufacturing line establishes a known population standard deviation (σ) of 12 grams. Prior to computing the test statistic, assumptions for a one-sample z-test must be satisfied. Random sampling is confirmed by the quality control protocol. Because the sample size n ≥ 30, the Central Limit Theorem dictates that the sampling distribution of the sample mean is approximately normal, fulfilling the distributional requirement (Triola, 2022).

Test Statistic Calculation

With a known population standard deviation and a sufficient sample size, a one-sample z-test provides the optimal evaluation metric. The test statistic utilizes the standard formula: z = (x̄ - μ) / (σ / √n). Substituting the observed values produces the following calculation:

x̄ = 495
μ = 500
σ = 12
n = 30

z = (495 - 500) / (12 / √30)
z = -5 / (12 / 5.477)
z = -5 / 2.19
z = -2.28
    

The resulting z-test statistic is -2.28. This value demonstrates that the sample mean rests 2.28 standard errors below the hypothesized population mean.

P-Value & Decision Rule

The alternative hypothesis specifies a left-tailed test (H_a: μ < 500). Consequently, the p-value represents the cumulative probability of observing a z-score less than or equal to -2.28 under the standard normal distribution. Utilizing standard statistical tables, the area to the left of -2.28 is 0.0113. The decision rule requires rejecting the null hypothesis if the calculated p-value falls below the significance level α = 0.05. Because 0.0113 < 0.05, the data mandate the rejection of the null hypothesis.

Conclusion in Context

At the 0.05 significance level, statistical evidence supports the rejection of the null hypothesis. The true population mean weight of the cereal boxes is significantly less than the claimed 500 grams. This 5-gram absolute deviation, combined with the low p-value (0.0113), indicates a systematic defect in the filling mechanism rather than random sampling variation. Immediate mechanical recalibration is recommended to prevent ongoing non-compliance with regulatory weight standards.

References

Moore, D. S., Notz, W. I., & Fligner, M. A. (2021). The Basic Practice of Statistics (9th ed.). W. H. Freeman.

Triola, M. F. (2022). Elementary Statistics (14th ed.). Pearson.