# Months for the results of our test of the bulbs

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months for the results of our test of the bulbs’ lifetime. So, we run our test of 1,000 light bulbs and determine the average simulated lifetime of the bulbs in months. We discover the following: Sample mean (written as x ) = 37.7 months Standard deviation (written as s ) = .42 months We use a t test in this case to determine if the null hypothesis should be rejected or ac- cepted (FTR) because we do not know the population’s standard deviation. However, be- cause the sample size is so large, the t value is the same as the Z value. A t test is constructed as follows: t = Samplevalue–Hypothesized population value Standa rd deviationof thesamplevalue In the Longlast light bulb case the calculation would be: t = - 37 7 36 0 42 4 05 . . . . See Table 13.3 for the appropriate statistical tests for the different forms of hypotheses. Step 5: Compare Results to the Null Hypothesis and Make a Decision Looking at a t distribution table we see that the t value for n-1 degrees of freedom (999 in our case, but “infinity” on the table) and for an alpha (the level of significance, which was .01 in our case) in a “one-tail” test (on a normal curve, the area under the curve at only one end of the curve rather than both ends) was 2.326. Since our t value of 4.05 exceeded this 2.326 value, we can say we are more than 99 percent confident that our sample mean of 37.7 months is drawn from a population whose mean is more than 36. Or, in other words, the chances that our sample mean would be 37.7 when the actual population mean is 36 months is very, very unlikely (less than one in a hun- dred). Because the odds of the null hypothesis being true (that our population mean is 36 months or less) are so small, it is much more reasonable to assume that the reason our sample mean was 37.7 months is that the null hypothesis should be rejected and our alternative hypothesis should be accepted (that the population mean is greater than 36 months).
Chapter 13 Advanced Data Analysis 229 In Step 1 we established a decision rule that said if we reject the null hypothesis and accept the alternative hypothesis we will market the Longlast light bulb, so that is what we should do. Referring to Table 13.2, this decision means we either will make the cor- rect decision (cell 4), or will be making a Type 1 error (cell 3). The actual chance that we are making a Type 1 error is called the “p value,” which in this case would be the likelihood of getting a sample mean of 37.7 months if the null hypothesis were true. This is very remote (less than 1 chance in 100), which is very reassuring of the correctness of our decision. While our null hypothesis was stated as a test of means (i.e., Was our sample’s aver- age of 37.7 months statistically significantly different from the hypothesized population’s average of 36 months?), we could state hypotheses as frequency distribution, differences between subgroups (e.g., men versus women), or proportions (e.g., percentages of college educated versus noncollege graduates intending to buy our product). Table 13.3 shows the appropriate statistical tests for different forms of hypotheses. Once we set up the hypoth-