Business Context

You’re the on-call data scientist supporting a high-throughput clinical diagnostics lab at a healthcare company processing ~80,000 patient samples/day. A routine immunoassay (used to quantify a biomarker that gates downstream clinical review) is monitored using daily quality-control (QC) runs. A “fail” triggers a production stop because biased results could lead to incorrect clinical decisions and regulatory exposure.

Yesterday, the assay failed QC: the control material (a standardized sample) produced unusually low readings. The lab team suspects either (a) a true shift in the assay calibration (systemic issue), or (b) a one-off problem like a bad reagent lot or pipetting error (localized issue). You have access to historical QC data and the last two days of runs.

Problem Statement

Using hypothesis testing and probability, determine whether the failure is more consistent with a systemic mean shift in the assay or a one-day anomaly, and quantify how surprising the observed QC result is under “business as usual.”

Given Data

The assay reports a continuous concentration value (units: ng/mL). Historically, the QC control is stable.

Operational detail: yesterday morning the lab switched to a new reagent lot on 6 of the 20 replicates (the other 14 used the old lot). You also have the replicate-level means by lot:

Requirements

1. Quantify surprise: Under the assumption the process is stable with mean μ₀ = 100 and σ = 4, compute the probability of observing a daily mean $\bar{X}\le 96.8$ with n = 20.
2. Hypothesis test (systemic shift): Perform a one-sided test for a downward shift in the mean using a t-test: $H_0: \mu = 100$ vs $H_1: \mu < 100$. Compute the test statistic and p-value.
3. Localize the root cause: Test whether the new reagent lot differs from the old lot using a two-sample Welch t-test (two-sided). Compute the test statistic and approximate p-value using the provided group means, group sizes, and assuming both groups have SD ≈ 4.2.
4. Construct a 95% confidence interval for the difference (new − old) and interpret it.
5. Provide a business recommendation: should the lab (i) stop the line and recalibrate immediately, (ii) quarantine the new lot and rerun QC, or (iii) treat as random noise?

Assumptions and Constraints

• Replicates within a day are approximately independent.
• For Requirement #1, treat σ = 4 as known and stable (historical estimate).
• For t-tests, assume approximate normality of replicate measurements (reasonable for aggregated assay noise).
• Ignore multiple-comparison corrections (single assay, single day), but discuss how you’d handle repeated daily monitoring in production.