For use only in [the name of your school] 2014 S4 Note. S4 Notes (Edexcel)

s (Edexcel) Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets 1

Quality of tests and estimators Type I and Type II errors. Size and Power of Test. The power test. Simple applications. Calculation of the probability of a Type I or Type II error. Use of Type I and Type II errors and power function to indicate effectiveness of statistical tests. Questions will not be restricted to the Normal distribution. Hypothesis Testing These have already been met in S2 and S3. When thinking about hypothesis tests it is clear that there are four possibilities: Actual Situation Conclusion that is made H 0 is true Accept H 0 H 0 is true Reject H 0 H 1 is true Accept H 0 H is true Reject H 0 1 Correct Decision Wrong Decision Wrong Decision Correct Decision So in two of above cases, the correct decision has been made and in two cases the wrong decision has been made. These wrong decisions are called errors and are given the names of Type I and Type II Errors, as explained below Type I and Type II Errors Actual Situation Conclusion that is made H 0 is true Accept H 0 H 0 is true Reject H 0 H 1 is true Reject H 1 H 1 is true Accept H 1 Correct Decision Wrong Decision Wrong Decision Correct Decision Type I Error Type II Error So it follows that: A Type I Error occurs if H 0 is rejected when H 0 is true. A Type II Error occurs if H 0 is accepted when H 0 is false. Use the above when asked for the definition in an exam question. However these can be easily remembered using the following (in the second statement all the numbers simply increase by 1): Type I Error) P(Rejecting H 0 H is true) P type I error P ( = 0. α is used to denote ( ) P(Type II Error) = P(Rejecting H H is true). β is used to denote P ( type II error ) 1 1 Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets 17

Type I Error Example A coin is tossed 8 times. p is the probability of the coin showing heads. It is wanted to test H 0 : p = 0. 5 against H : 0.5 1 p < at a 5% significance level. Find the probability of making a Type I Error. A helpful starting point is always to write down P( Type I Error) = P( rejecting Ho Ho is true) So, in this example P Type I Error = P rejecting H p = 0.5 ( ) ( o ) Let X be the random variable the number of heads when the coin is tossed 8 times. Assume H 0 and so X ~ B (8, 0.5). From the tables, PX ( 1) = 0.0352 and PX ( 2) = 0.1445. So, if X = 2, since PX ( 2) > 5%, H 0 would not be rejected. But, if X 1, then, since PX ( 1) < 5%, H 0 would be rejected. So, ( ) ( X p ) P Type I Error = P 1 = 0.5 = 0.0352 Example The lengths of metal bars produced by a particular machine are normally distributed with mean length 420cm and standard deviation 12cm. The machine is serviced, after which a sample of 100 bars is chosen. It is wanted to test H 0 : µ = 420cm (there is no change) against H 1 : µ 420cm (there is a change) at a 5% significance level. Find the probability of making a Type I Error. A helpful starting point is always to write down P( Type I Error) P( rejecting Ho Ho is true) So, in this example P Type I Error = P rejecting H µ = 420 ( ) ( o ) Assuming H 0, 2 12 X ~ N 420,. 100 =. Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets 18

By definition, in this case, P( Type I Error) = 5% since H0 is rejected in the region which has exactly 2.5% at either end of the distribution. So If the data is continuous, for example if it was normally distributed, then P(Type I Error) = significance level of test. If the data is discrete then P(Type I Error) < significance level of test. Type II Error As was stated earlier, a type II error occurs when H 0 is not rejected when H 1 is true. P type II error = P H is rejected H is true So ( ) ( 1 1 ) β is used to denote P ( type II error) and it cannot be calculated without further information. The steps are as follows: STEP 1 Find the critical region for which H 0 is rejected. STEP 2 Find the probability of not being in this critical region, using the parameter that has been given. Example A man claims that 40% of his candidates pass the exam. 12 candidates are chosen at random. It is wanted to test H 0 : His claim is correct against H 1 : His claim is an underestimate at a 5% significance level. Find the probability of making a Type II Error given that it was later found that 60% of his candidates pass. P type II error = P H is rejected H is true A helpful starting point is always to write down ( ) ( 1 1 ) STEP 1 Find the critical value for which H 0 is rejected. ( 8) = 1 P( X 7) P X = 1 0.9427 = 0.0573 ( 9) = 1 P( X 8) P X = 1 0.9847 = 0.0153 Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets 19

P X 9 < 5%. The critical region is X 9. So, since ( 8) 5% P X > and ( ) STEP 2 Find the probability of not being in this critical region, using the parameter that has been given. So P type II error = P X 8 p= 0.6 ( ) ( ) Let Y be the number of people who fail, X Y 12 ( type II error) = ( 8 = 0.6) = PY ( 4 q= 0.4) = 1 PY ( 3 q= 0.4) P P X p = 1 0.2253 = 0.7747 + = and Y ~ B ( 12, 0.4) Copyright www.pgmaths.co.uk - For AS, A2 notes and IGCSE / GCSE worksheets 20