Construction defects cases are those brought by property owners who allege that the construction of their homes fail to meet the contractual obligations of the builder. One (2019) case* involved a Florida community consisting of 276 homes. Allegations of faulty stucco installation on the homes were brought by 57 plaintiff homeowners. The plaintiffs hired an engineering firm to test the stucco. The firm sampled 53 homes for testing. The defendant construction firm hired a statistical consultant to determine whether the sample was representative of the population of 276 homes. She found that 51 of the 53 homes were owned by plaintiff homeowners.

a. Use statistical software to find the probability that a random sample of 53 homes from the population of 276 homes would result in at least 51 homes owned by plaintiffs.

b. Refer to part a. The statistical consultant concluded that the sample selection was biased toward over-inclusion of plaintiff homes, and that the engineering firm’s inferences about all 276 homes based on this sample were likely to be biased. Do you agree? Explain.

c. Once the homes were selected, samples of 12-inch by 12-inch sections of stucco were selected for testing. A total of 98 sections were selected from the 53 homes. The statistician learned that 79 of the 98 sections bordered window openings. Use statistical software (and the hint in part a) to find the probability of randomly sampling at least 79 sections bordering window openings, assuming (as did the statistician) that as many as 50% of all possible sections bordered windows.

d. Refer to part b. The statistical consultant concluded that in addition to the homes sampled being biased toward the selection of plaintiff homes, the stucco sections were biased toward window openings. Ultimately, she reported to the Court that the severe bias in the sample selections of homes and stucco sections made any inference about the stucco on all 276 homes unreliable. Do you agree? Explain.