# Errors in Hypothesis Testing

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06-Jan-2016Category

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Errors inHypothesis Testing

2 TYPES OF ERRORS TRUE CASE HAis true HAis false

WE Accept HA

SAY Do not Accept HA

CORRECTCORRECTTYPE IERRORTYPE IIERRORPROB = PROB = is set by the decision maker varies and depends on:(1) ; (2) n; (3) the true value of

Relationship Between and is the Probability of making a Type II errori.e. the probability of not concluding HA is true when it is depends on the true value of The closer the true value of is to its hypothesized value, the more likely we are of not concluding that HA is true -- i.e. is large (closer to 1)

is calculated BEFORE a sample is takenWe do not use the results of a sample to calculate

CALCULATING Example: If we take a sample of n = 49, with = 4.2, What is the probability we will get a sample from which we would not conclude > 25 when really = 25.5? (Use = .05)REWRITE REJECTION REGION IN TERMS OF

CALCULATING (contd)So when = 25.5,If we get an > 25.987, we will correctly conclude that > 25If we get an < 25.987 we will not conclude that > 25 even though really = 25.5

CALCULATING (contd)So what is P(not getting an > 25.987 when really = 25.5? That is P(getting an < 25.987)?Calculate z = (25.987 - 25.5)/(4.2/ ) .81 is the area to the left of .81 for a > testP(Z < .81) = .7910

> TestDetermining When = 25.50 .81 Z25.5 25.987 .7910

What is When = 27?So what is P(not getting an > 25.987 when really = 27? That is P(getting an < 25.987)?Calculate z = (25.987 - 27)/(4.2/ ) -1.69 is the area to the left of -1.69 for a > testP(Z < -1.69) = .0455

This shows that the further the true value of is from thehypothesized value of , the smaller the value of ; that is weare less likely to NOT conclude that HA is true (and it is!)

> TestDetermining When = 27-1.69 0 Z25.987 27 .0455

- for
What is When = 25.5?So what is P(not getting an < 26.013 when really = 25.5? That is P(getting an > 26.013)?Calculate z = (26.013 25.5)/(4.2/ ) .86 is the area to the right of .86 for a .86) = 1 - .8051 = .1949

for TestsFor n = 49, = 4.2, What is the probability of not concluding that 26, when really is 25.5? (With = .05)

This time is the area in the middle between the two critical values of

What is When = 25.5?So what is P(not getting an < 24.824 or > 27.176 when really = 25.5? That is P(24.824 < < 27.176)?Calculate zs = (24.824 25.5)/(4.2/ ) -1.13 and = (27.176 25.5)/(4.2/ ) 2.79 is the area in between -1.13 and 2.79 for a testP(Z < 2.79) = .9974 P(Z < -1.13) = .1292 P(-1.13 < Z < 2.79 = .9974 - .1292 = .8682

TESTDetermining When = 25.5 -1.13 0 2.79 Z24.824 25.5 27.176

The Power of a Test = 1 - is the Probability of making a Type II errori.e. the probability of not concluding HA is true when it is depends on the true value of and sample size, nThe Power of the test for a particular value of is defined to be the probability of concluding HA is true when it is -- i.e. 1 -

Power Curve CharacteristicsThe power increases with:Sample Size, nThe distance the true value of is from the hypothesized value of

Power Curves For HA: 26With n = 25 and n = 49

Calculating Using Excel> TestsSuppose H0 is = 25; = 4.2, n = 49, = .05> TESTS: HA: > 25 and we want when the true value of = 25.5

1)Calculate the critical x-bar value = 25 + NORMSINV(.95)*(4.2/SQRT(49))2) Calculate z = (critical x-bar -25.5)/ (4.2/SQRT(49))3) Calculate the the probability of getting a z- value < than this critical z value: -- this is =NORMSDIST(z)

Calculating Using Excel< TestsSuppose H0 is = 27; = 4.2, n = 49, = .05< TESTS: HA: < 27 and we want when the true value of = 25.5

1)Calculate the critical x-bar value = 27 - NORMSINV(.95)*(4.2/SQRT(49))2) Calculate z = (critical x-bar -25.5)/ (4.2/SQRT(49))3) Calculate the the probability of getting a z- value > than the critical value: -- this is =1-NORMSDIST(z)

Calculating Using Excel TestsSuppose H0 is = 26; = 4.2, n = 49, = .05 TESTS: HA: 26 and we want when the true value of = 25.5 1)Calculate the critical upper x-barU value and the lower critical x-barL value = 26 - NORMSINV(.975)*(4.2/SQRT(49)) (x-barL)= 26 + NORMSINV(.975)*(4.2/SQRT(49)) (x-barU)2) Calculate zU = (x-barU -25.5)/ (4.2/SQRT(49)) and zL = (x-barL -25.5)/ (4.2/SQRT(49)) 3) Calculate the the probability of getting an z- value in between zL and zU - this is =NORMSDIST(zU) - NORMSDIST(zL)

for > Tests

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for Tests

- REVIEWType I and Type II Errors = Prob (Type I error) = Prob (Type II error) -- depends on , n and How to calculate for:> Tests

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