3. one-tailed tests

in a one-tailed hypothesis test, the (one-tailed) alternative hypothesis is tested to check if it is correct

testing the alternative hypothesis when it is greater than the null hypothesis

a test statistic is introduced, 10 events are observed, the null hypothesis assumes there is a 0.4 probability of success, the alternative hypothesis assumes there is a greater than 0.4 probability of success because a test has obtained 9 successes, the significance level is 5%

H0 : p = 0.4
H1 : p > 0.4

X~B(10, 0.4)

the critical value is x = 8, the critical region is x => 8, the acceptance region is x <= 7

9 falls inside the critical region, so the null hypothesis would be rejected

1 - P(X <= 8) = P(X => 9)

1 - 0.998 = 0.002

0.002 < 0.05

so the null hypothesis would be rejected

testing the alternative hypothesis when it is less than the null hypothesis

a test statistic is introduced, 10 events are observed, the null hypothesis assumes there is a 0.4 probability of success, the alternative hypothesis assumes there is a less than 0.4 probability of success because a test has obtained 2 successes, the significance level is 5%

H0 : p = 0.4
H1 : p < 0.4

X~B(10, 0.4)

the critical value is x = 1, the critical region is x <= 1, the acceptance region is x => 2

2 falls outside the critical region, so the null hypothesis would be accepted

P(X <= 2) = 0.167

0.167 > 0.05

so the null hypothesis would be accepted