4. two-tailed tests

a one-tailed test is used to test the claim that the probability has either gone up or gone down

a two-tailed test is used to test the claim that the probability could have changed in either direction

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

testing the alternative hypothesis when the outcome is less than the expected outcome

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 not a 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 expected outcome is np, 4

2 is less than 4, so test P(X <= 2)

P(X <= 2) = 0.167

test against half the significance level (because it is a two-tailed test)

0.167 > 0.025

so the null hypothesis would be accepted

the two critical regions → x = 0 AND x => 7

2 falls outside the critical region

0 =/= 2 AND 2 < 7

so the null hypothesis would be accepted

testing the alternative hypothesis when the outcome is greater than the expected outcome

H0 : p = 0.4

H1 : p =/= 0.4

X~B(10, 0.4)

the expected outcome is np, 4

9 is greater than 4, so test P(X => 9)

1 - P(X <= 8) = P(X => 9)
1 - 0.998 = 0.002

test against half the significance level (because it is a two-tailed test)

0.002 < 0.025

so the null hypothesis would be rejected

the two critical regions → x = 0 AND x => 7

9 falls within one critical region

9 > 7

so the null hypothesis would be rejected