A tyre company claims that the mean life of its tyres is at least 40000 km. A consumer group suspects that the mean life is less than this. If \(\mu\) is the mean life (in km) of the tyres, the appropriate null and alternative hypotheses are
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(H_0: \mu \ge 40000\), \(H_1: \mu \lt 40000\)
The null hypothesis \(H_0\) is the claim being tested and always carries the equality sign (=, ≤ or ≥). The company’s claim “at least 40000 km” gives
\(H_0: \mu \ge 40000\)
The alternative hypothesis \(H_1\) is the opposite view, which here is the consumer group’s suspicion “less than 40000 km”:
\(H_1: \mu \lt 40000\)
Why the other options are incorrect
Option (1) \(H_0: \mu \le 40000\), \(H_1: \mu \gt 40000\) — Incorrect. This tests whether the life is more than 40000 km, but the suspicion is that it is less.
Option (2) \(H_0: \mu \ne 40000\), \(H_1: \mu = 40000\) — Incorrect. The equality sign must be in \(H_0\), not in \(H_1\).
Option (4) \(H_0: \mu \lt 40000\), \(H_1: \mu \ge 40000\) — Incorrect. The two hypotheses are swapped: the company’s claim (with the equality sign) belongs in \(H_0\).