Answer:
Step-by-step explanation:
a) We would set up the hypothesis test.
For the null hypothesis,
µ ≤ 2
For the alternative hypothesis,
µ > 2
b) at 5% level, α = 0.05, the critical value is determined from the normal distribution table.
For the left, α/2 = 0.05/2 = 0.025
The z score for an area to the left of 0.025 is - 1.96
For the right, α/2 = 1 - 0.025 = 0.975
The z score for an area to the right of 0.975 is 1.96
The rejection regions are z = 1.96 and z = - 1.96
c) the population standard deviation is known and the sample size is large. if the sample mean purity, x = 1.82, then we would determine the z score by applying the formula
z = (x - µ)/(σ/√n)
Where
x = sample mean
µ = population mean
σ = population standard deviation
n = number of samples
From the information given,
µ = 2
x = 1.82
σ = 0.5
n = 75
z = (1.82 - 2)/(0.5/√75) = - 3.12
Since α = 0.01, the critical value is determined from the normal distribution table.
For the left, α/2 = 0.01/2 = 0.005
The z score for an area to the left of 0.005 is - 2.575
For the right, α/2 = 1 - 0.005 = 0.995
The z score for an area to the right of 0.995 is 2.575
In order to reject the null hypothesis, the test statistic must be smaller than - 2.575 or greater than 2.575. Since - 3.12 < - 2.575 and 3.12 > 2.575, we would to reject the null hypothesis.
d) if critical point is 1.9, we would determine α
The z probability value to 1.9 on the right is 0.971. For the right,
α/2 = 1 - x = 0.971
α/2 = 1 - 0.971 = x
α/2 = 1 - 0.029 = x
α = 2 × 0.029
α = 0.058
