Question

1.       Write the decision rule for the following tests:

a.       H₀: µ = 75;    H₁: µ ≠ 75                    σ = 5                      α = 0.05                n = 100 x=75

b.       H₀: µ ≤ 75;   H₁: µ > 75                    σ = 5                      α = 0.05                n = 100 x=75

c.       H₀: µ ≥ 75;    H₁: µ < 75                    s = 5                       α = 0.05                n = 20 x=75
(note that s is the sample standard deviation)

d.       H₀: µ = 75;    H₁: µ ≠ 75                    s = 5                       α = 0.05                n = 20 x=75

e.       H₀: π ≤ 0.60;   H₁: π > 0.60             α = 0.01                n = 200 x=0.60

f.        H₀: π = 0.60;   H₁: π ≠ 0.60             α = 0.01                n = 200 x=0.60

Answer 1

Solution:

Write the decision rule for the following tests:

a. H0: µ = 75;    H1: µ ≠ 75                    σ = 5                      α = 0.05                n = 100 x=75

Answer: Here we have a two-tailed alternative hypothesis and population standard deviation is known, we, therefore, have to find two-tailed z critical values based on normal distribution at significance level 0.05/2 and they are:

Therefore the decision rule is:

Reject the null hypothesis, if the test-statistic, or

b.       H0: µ ≤ 75;   H1: µ > 75                    σ = 5                      α = 0.05                n = 100 x=75

Answer: Here we have a right-tailed alternative hypothesis and population standard deviation is known, we, therefore, have to find right-tailed z critical value based on normal distribution at significance level 0.05 and it is:

Therefore the decision rule is:

Reject the null hypothesis, if the test statistic,

c.       H0: µ ≥ 75;    H1: µ < 75                    s = 5                       α = 0.05                n = 20 x=75

Answer: Here we have a left-tailed alternative hypothesis and population standard deviation is unknown with sample size less than 30, we, therefore, have to find left-tailed t critical value based on t distribution at significance level 0.05 for and it is:

Therefore the decision rule is:

Reject the null hypothesis, if the test statistic,

d.       H0: µ = 75;    H1: µ ≠ 75                    s = 5                       α = 0.05                n = 20 x=75

Answer: Here we have a two-tailed alternative hypothesis and population standard deviation is unknown with sample size less than 30, we, therefore, have to find two-tailed t critical values based on t distribution at significance level 0.05/2 for and they are:

Therefore the decision rule is:

Reject the null hypothesis, if the test-statistic, or

e.       H0: π ≤ 0.60;   H1: π > 0.60             α = 0.01                n = 200 x=0.60

Answer: Here we have a right-tailed alternative hypothesis, we, therefore, have to find the right-tailed z critical value based on normal distribution at significance level 0.01 and it is:

Therefore the decision rule is:

Reject the null hypothesis, if the test statistic,

f.        H0: π = 0.60;   H1: π ≠ 0.60             α = 0.01                n = 200 x=0.60​

Answer: Here we have a two-tailed alternative hypothesis, we, therefore, have to find two-tailed z critical values based on normal distribution at significance level 0.01/2 and they are:

Therefore the decision rule is:

Reject the null hypothesis, if the test-statistic, or