EXAMPLE 93

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Germinating Bulbs

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A horticulturist is experimenting with an altered bulb for a large plant From previous experience, she knows that the percentage of these bulbs that germinate is either 50% or 75% To decide which germination rate is correct, she plans an experiment involving 15 of these altered bulbs and records the number of bulbs that germinate We assume that the number of bulbs that germinate follows a binomial model, that is, a bulb either germinates or it does not, the bulbs behave independently, and the probability of germination is constant If in fact the probability is 50% that a bulb germinates and if X is the random variable denoting the number of bulbs that germinate, then P(X = x) = 15 (050)x (050)15 x x for x = 0, 1, 2, , 15

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9

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Statistical Inference I

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while if the probability is 75% that a bulb germinates, then P(X = x) = 15 (075)x (025)15 x x for x = 0, 1, 2, , 15

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We should rst consider the probabilities of all the possible outcomes from the experiment These are shown in Table 91 Table 91 Probabilities for Example 93 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 p = 050 00000 00004 00032 00139 00417 00916 01527 01964 01964 01527 00916 00417 00139 00032 00004 00000 p = 075 00000 00000 00000 00000 00001 00007 00034 00131 00393 00917 01651 02252 02252 01559 00668 00134

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The statements that 50% of the bulbs germinate or 75% of the bulbs germinate are called hypotheses They are conjectures about the behavior of the bulbs We will formalize these hypotheses as H0 : p = 050 Ha : p = 075 We have called H0 : p = 050 the null hypothesis and Ha : p = 075 the alternative hypothesis Now we must decide between them If we decide that the null hypothesis is correct, then we accept the null hypothesis and reject the alternative hypothesis On the contrary, if we reject the null hypothesis then we accept the alternative hypothesis How should we decide The decision process is called hypothesis testing In this case, we would certainly look at the number of bulbs that germinate If in fact 75% of the bulbs germinate, then we would expect a large number of the bulbs to germinate It would appear, if a large number of bulbs germinate, say 11 or more, that we would then reject the null hypothesis (that p = 050) and accept the alternative hypothesis (that p = 075) In coming to this test, we cannot reach a decision with certainty because our conclusion is based on a sample, a small one at that in this instance What are the risks involved There are two risks or errors that we can make: we could reject the null hypothesis when it is actually true or we could accept the null hypothesis when it is false Let us consider each of these

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Hypothesis Testing

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Rejecting the null hypothesis when it is true is called a Type I error In this case, we reject the null hypothesis when the number of germinating bulbs is 11 or more The probability this occurs when the null hypothesis is true is P(Type I error) = 00417 + 00139 + 00032 + 00004 + 00000 = 00592 So about 6% of the time, bulbs that have a germination rate of 50% will behave as if the germination rate were 75% Accepting the null hypothesis when it is false is called a Type II error In this case, we accept the null hypothesis when the number of germinating bulbs is 10 or less The probability this occurs when the null hypothesis is false is P(Type II error) = 00000 + 00000 + 00000 + 00000 + 00001 + 00007 + 00033 + 00131 + 00393 + 00917 + 01651 = 03133 So about 31% of the time, bulbs with a germination rate of 75% will behave as though the germination rate were only 50% The experiment will always result in some value of X We must decide in advance which values of X cause us to accept the null hypothesis and which values of X cause us to reject the null hypothesis The values of X that cause us to reject H0 comprise what we call the critical region for the test In this case, large values of X are more likely to come from a distribution with p = 075 than from a distribution with p = 050 We have used the critical region X 11 here So it is reasonable to conclude that if X 11, then p = 075 The errors calculated above are usually denoted by and In general then = P(H0 is rejected if it is true) where is often called the size or the signi cance level of the test The size of the Type II error is denoted by In general then = P(H0 is accepted if it is false) In this case, with the critical region X 11, we nd = 00542 and = 03133 Note that and are calculated under quite different assumptions, since presumes the null hypothesis true and presumes the null hypothesis false, so they bear no particular relationship to one another It is of course possible to decrease by reducing the critical region to say X 12 This produces = 00175, but unfortunately, the Type II error increases to 05385 The only way to decrease both and simultaneously is to increase the sample size Finally note that both and increase or decrease in nite amounts It is not possible to nd a critical region that would produce between the values 00175 and 00542 It is possible to decrease both and by increasing the sample size as we now show

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