| Test rejects Null | Test fails to reject Null | |
|---|---|---|
| Null is true | Type 1 error- false positive | Correct decision- no effect |
| Null is false | Correct decision- no effect | Type 2 error- false negative |
Lucy can hit the target 70% of the time when she throws an axe with her right hand. She claims that the proportion, p, of her throws that hit the target is higher than 70% when she uses her left hand. Lucy uses the hypotheses \(H_{0}:p = 0.7\) and \(H_{1}:p > 0.7\) to test her claim. Lucy makes 100 throws and will reject the null hypothesis if the axe hits the target more than 77 times. Let X -B(100,p) be the number of times lucy hits the target when using her left hand, P(Type 1 error) = P(being in critical region|Ho is true) P(Type 1 error) = \(P(X \gt 77| p = 0.7)\) P(Type 1 error) = \(P(78\leq X \leq 100|p = 0.7)=0.04786\)
Lucy can hit the target 70% of the time when she throws an axe with her right hand. She claims that the proportion, p, of her throws that hit the target is higher than 70% when she uses her left hand. Lucy uses the hypotheses \(H_{0}:p = 0.7\) and \(H_{1}:p > 0.7\) to test her claim. Lucy makes 100 throws and will reject the null hypothesis if the axe hits the target more than 77 times. Let X -B(100,p) be the number of times lucy hits the target when using her left hand. Given that Lucy actually hits the target 80% of the time with her left hand, find the probability of a Type II error P(Type II error) = P(not being in critical region|true population parameter) P(Type II error) = \(P(X \leq 77| p = 0.8)\) P(Type II error) = \(P(0\leq X \leq 77| p = 0.8)\) P(Type II error) = 0.261
In the following scenarios, decide whether a Type I error or Type II error could have occurred (i) A farmer is testing for a change in crop growth after trying a new fertiliser. The test concludes that there is no evidence of change at the 5% significance level. (ii) A dentist’s receptionist believes that the waiting times have been reduced due to a new scheduling system. They conduct a hypothesis test and will reject the null hypothesis if no more than two customers wait more than ten minutes. Exactly two customers have to wait more than ten minutes.
The probability of getting a head when a coin is tossed is denoted by p. This coin is tossed 12 times in order to test the hypotheses H0: p = 0.5 against H1: \(p \neq 0.5\), using a 5% level of significance. Given that p = 0.4 Find the probability of a type-2 error. When p = 0.5 rejection region \(P(X\geq 10)\) or \(P(X\leq 2)\) When p = 0.4 acceptance region \(P(3\geq X\geq 9)= 0.9137\)