# ch6 - Stat 0302B Business Statistics Spring 2010-2011...

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Stat 0302B Business Statistics Spring 2010-2011 Chapter VI Basic Concepts of Testing Hypothesis § 6.1 Introduction In estimation, we start with a sample statistic and make a statement about the population parameter. In hypothesis testing, we start by assuming a statement about the population (usually in terms of some parameters) and argue whether this statement was supported by the data. Hypothesis A statistical hypothesis is an assertion or statement about the population, usually formulated in terms of population parameters. It is denoted by or . 0 H 1 H Null Hypothesis The null hypothesis , denoted by , is usually a statement about something that 0 H has been established, or something that has an authoritative standing, or something worth protecting. Alternative Hypothesis The alternative hypothesis , denoted by , is usually a statement about something 1 H that challenges the authority, or something that needs not be protected strongly. The hypotheses are often expressed in terms of the population parameters. However, there are exceptional cases in non-parametric analysis. Example 6.1 = 118 : 118 : 1 0 μ H H , , = 1 : 1 : 2 1 2 0 σ H H = 2 1 1 2 1 0 2 : 2 : p p H p p H on. distributi normal not is Population : on. distributi normal is Population : 1 0 H H P.119

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Stat 0302B Business Statistics Spring 2010-2011 Example 6.2 An intensive survey conducted a few years ago found that the mean IQ of HKU students was 118. Some people suspect that the mean IQ has decreased recently. If a random sample of 25 students gives an average IQ score 114.3, which claim should we take? Assume that the population standard deviation IQ is 15 = σ . Null hypothesis: 118 : 0 = μ H Alternative hypothesis: 118 : 1 < H The basic strategy in hypothesis testing is to measure how far an observed statistic is from a hypothesized value of the parameter. If the distance is “great”, we would argue that the hypothesized statement is inconsistent with the data and we would be inclined to reject the hypothesis. (We could be wrong, of course; rare events do happen!) distance 118 = 3 . 114 = X As a first glance, the data seems to support the alternative hypothesis 118 : 1 < H and opposite the null hypothesis 118 : 0 = H . However, before jumping to the conclusion, we must take into account the possible variability of the observations as the distance may be just resulted from sampling errors but not the deviation of the null hypothesis from the truth. A strong conclusion may be drawn if the distance is large enough. As a common practice, this distance may be compared with 2 standard errors of the mean. Standard error of X : 3 25 15 = = SE 2 × SE = 6 3 . 114 = X 118 = 120.3 P.120
Stat 0302B Business Statistics Spring 2010-2011 We had used 2 SE to take into account the possible variability due to sampling errors. Since the observed sample mean is smaller than the hypothetical mean 118 by less than 2 SE , it is quite possible to observe this small sample mean even though the population mean IQ score is 118. Therefore the data didn’t provide enough evidence for us to conclude that the hypothesis 118 : 0 = μ H is not true.

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ch6 - Stat 0302B Business Statistics Spring 2010-2011...

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