2. finding critical values

the critical region = the region of values where, if the test statistic falls within, would cause the null hypothesis to be rejected

the critical value = the value that seperates the critical region from the acceptance region

finding the critical value in a one-tailed test where the alternative hypothesis 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, the significance level is 5%

  • H0 : p = 0.4

  • H1 : p > 0.4

  • X~B(10, 0.4)

the probability of there being 8 or more successes

  • P(X => 8) = 0.012

the probability of there being 7 or more successes

  • P(X => 7) = 0.055

so, a test statistic of 8 or more would lead to the null hypothesis being rejected, and a test statistic of 7 or less would lead to the null hypothesis being accepted

the critical value → x = 8

the critical region → x => 8

the acceptance region → x <= 7

the actual significance level = 1.2%

the probability of incorrectly rejecting the null hypothesis = 1.2%

finding the critical value in a one-tailed test where the alternative hypothesis 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, the significance level is 5%

  • H0 : p = 0.4

  • H1 : p < 0.4

  • X~B(10, 0.4)

the probability of there being 1 or less successes

  • P(X <= 1) = 0.046

the probability of there being 2 or less successes

  • P(X <= 2) = 0.167

0.046 < 0.05

so, a test statistic of 1 or less would lead to the null hypothesis being rejected, and a test statistic of 2 or more would lead to the null hypothesis being accepted

the critical value → x = 1

the critical region → x <= 1

the acceptance region → x => 2

the actual significance level = 4.6%

the probability of incorrectly rejecting the null hypothesis = 4.6%

two-tailed test