Question

A major cause of injuries in highway work zones is weak construction barriers. One federal government road-barrier specification involves the velocity of a front-seat passenger immediately following impact. This mean impact velocity must be less than 12 m/s. TSS GmbH, a German company, has just developed a new high-impact road barrier with special absorbing material. In controlled tests the impact velocity was measured for 9 randomly selected crashes. The sample mean was 11.85 m/s. Assume the distribution of impact velocities is normal and σ = 0.56. Please use 3 decimal places in all calculations.

a) State the hypotheses that are being tested for the study that the impact velocity is acceptable to the federal government. Be sure to include both the null and alternative hypotheses.

b) Find the probability of the Type II error if the true mean impact velocity is μ_a = 11.6; that is, find β(11.6). Assume that α = 0.05.

Answer 1

a) H₀: > 12

   H₁: < 12

b) At = 0.05 the critical value is z_crit = -1.645

z_crit = -1.645

or, ( - )/() = -1.645

or, ( - 12)/(0.56/) = -1.645

or, ( - 12) = -1.645 * (0.56/)

or, = -1.645 * (0.56/) + 12

or, = 11.693

= P( > 11.693)

               = P(( - )/() > (11.693 - )/())

               = P(Z > (11.693 - 11.6)/(0.56/))

                = P(Z > 0.50)

                = 1 - P(Z < 0.50)

                = 1 - 0.6915

                = 0.3085