Form and interpret hypothesis tests for a population mean

Unit 4 modelling task 2: test a specified prediction

At T=2.5ln3 days the logistic model predicts 40 biomass units. For this exercise, 64 independently prepared cultures are a random sample under the same target conditions; their endpoint observations are iid normal with known population SD 4 units. Their mean is 41.2. Before data, choose H₀:μ=40 versus H₁:μ≠40 at 5%. The candidate parameters are specified independently; reusing these data to fit them would require a different analysis

For 64 independent observations with known population SD 4 biomass units, what is the standard error of the sample mean in biomass units?

For an observed sample mean of 41.2, null mean 40 and standard error 0.5, what is the observed z statistic?

Write and critique a complete validation report, ungraded

The model predicts an endpoint population mean of 40 biomass units under the target conditions. There are 64 independent, randomly sampled endpoint observations with mean 41.2 and known population SD 4 biomass units; the two-sided test was specified at 5% before data. Calculate the two-sided p-value and a 95% known-sigma confidence interval for μ. Interpret the test, interval and observed difference 1.2 in context. A practical tolerance of 2 units was specified before data. Explain whether the result establishes an error exceeding that tolerance, and identify limits of inference about the full dynamic model

Model: report evidence and its precise scope

For a prespecified two-sided 5% test of a predicted endpoint mean of 40 biomass units, 64 independent observations have mean 41.2 and known population SD 4 biomass units. The two-sided p-value is2Φ(−2.4)≈0.0164, so reject μ=40 at 5% under the stated sampling model. A95% interval is 41.2±1.9599639845…×0.5, or approximately[40.2200,42.1800]. It excludes 40, agreeing with the two-sided 5% test. The observed mean difference is 1.2 units; the corresponding difference interval is approximately[0.2200,2.1800]. This does not establish that the discrepancy exceeds 2 units, nor establish that it stays within 2. The test concerns the endpoint population mean under these conditions, not every observation or all times. Check measurement calibration, independence and representativeness before attributing the discrepancy to the differential equation

For a two-sided z test with observed z = 2.4, give the p-value to four decimal places

The 95% interval for the mean difference is approximately [0.2200, 2.1800] units. Which conclusion about the prespecified 2-unit tolerance follows?

Technology-enabled exam practice: document the statistical commands

In a statistical tool, evaluate 2×normalCDF(−2.4) for the standard normal and invNormal(0.975) for its critical quantile; check whether your tool expects a lower-tail probability and standard deviation or variance. Retain unrounded values internally: the sample size is 64, the known population SD is 4 biomass units, and μ is the endpoint population mean under the target conditions. The normal-CDF value is a tail probability under H₀, not a probability that H₀ is true

Vary the sampling assumptions

If the 64 values were repeated readings on one culture, independence and population representation would need justification; σ/√64 cannot simply be asserted. If σ were unknown, substituting a sample SD would give an approximate normal-reference analysis only when justified here. If the model parameters were fitted to the same sample, do not use this fixed-prediction test unchanged

Human marking guide for the statistical task

Look for the named target mean, prespecified hypotheses and level, correct SE/statistic/two tails, justified decision, correct interval interpretation, practical threshold comparison and limitations