# Will be rejected only if the sample evidence strongly

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will be rejected only if the sample evidence strongly suggests that H 0 is false. In this case H a must be true. Otherwise H 0 will not be rejected. So there are two possible conclusions: reject H 0 (accept H a ) do not reject H 0 7
Note that these decisions are not symmetric, there is no way you can say you accept H 0 . Remark: Hypotheses should be the logical complement of each other. Common choices of hypotheses are H 0 : population characteristic = specific value versus H a : population characteristic ̸ = specific value H 0 : population characteristic specific value versus H a : population characteristic > specific value H 0 : population characteristic specific value versus H a : population characteristic < specific value Example 3 H 0 : p = 0 . 25 versus H a : p ̸ = 0 . 25 H 0 : μ 100 versus H a : μ < 100 We can’t test H 0 : μ 100 versus H a : μ > 150 Be careful when choosing hypotheses, because a statistical test can only support the alternative hypothesis, by rejecting H 0 . Is H 0 not being rejected does not mean strong support for H 0 , but lack of strong evidence against H 0 . Example 4 A company is advertising that the average lifetime of their light bulbs is 1000 hours. You might question this, and want to show that in fact the lifetime is shorter. Let μ = mean lifetime of the light bulbs. You would test H 0 : μ 1000 versus H a : μ < 1000. Rejection of H 0 would then support your claim, that the mean lifetime is less than 1000 hours. However, non-rejection of H 0 does not necessarily provide strong support for the advertised claim, that the mean lifetime is at least 1000 hours. How to make a decision (reject H 0 , or do not reject H 0 ) Since we do Statistics the decision to reject, or not to reject H 0 is based on information contained in a sample drawn from the population of interest. We use the sample to 1. Calculate a test statistic,
2. Evaluate the value of the test statistic based on its distribution under the assumption that H 0 is true. 8
3. Make a decision: Is the value of the test statistic highly unlikely to occur, under the assumption that H 0 is true, we will interpret this as a contradiction to the assumption that H 0 is true, and reject this hypothesis and decide that H a must be true.
Example 5 Suppose μ is the mean in a given population which follows a normal distribution with standard deviation σ = 0 . 2. The investigator wants to test H 0 : μ 0 . 5 versus H a : μ < 0 . 5. A random sample of size 100 showed a sample mean of ¯ x = 0 . 3. which is less than 0 . 5, as claimed in the alternative hypothesis.