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The One-Tailed t-Test As usual buy discount beconase aq on-line, we perform one-tailed tests when we predict the direction of the difference between our conditions order beconase aq paypal. Thus 200MDI beconase aq amex, if we had predicted that men score higher than women Ha would be that the sample represents a population with greater than 75 1Ha: 7 752. We then examine the sampling distribution that occurs when 5 75 (as we did in the two- tailed test). To decide in which tail of the sampling distribution to put the region of rejection, we determine what’s needed to support Ha. Here, for the sample to represent a population of higher scores, the X must be greater than 75 and be significant. However, predicting that men score lower than women would produce the sampling distribution on the right in Figure 11. Because we seek a X that is significant and lower than 75, the region of rejection is in the lower tail, and tcrit is negative. If the absolute value of tobt is larger than tcrit and has the same sign, then the X is unlikely to be representing a described by H0. When the df of your sample does not appear in the table, you can take one of two approaches. First, remember that all you need to know is whether tobt is beyond tcrit, but you do not need to know how far beyond it is. Often you can determine this by examining the critical values given in the table for the df immediately above and below your df. The second approach is used when tobt falls between the two critical values given in the tables. Then you must compute the exact tcrit by performing “linear interpolation,” as described in Appendix A. X (continued) Estimating by Computing a Confidence Interval 243 For Practice Answers 1. Conclusion: Artificial sunlight signif- obt crit icantly lowers depression scores from a of 8 to a 1. The first way is point estimation, in which we describe a point on the variable at which the is expected to fall. Earlier we estimated that the of the population of men is located on the variable of housekeeping scores at the point identified as 65. How- ever, if we actually tested the entire population, would probably not be exactly 65. The problem with point estimation is that it is extremely vulnerable to sampling error. Our sample probably does not perfectly represent the population of men, so we can say only that the is around 65. The other, better way to estimate is to include the possibility of sampling error and perform interval estimation. With interval estimation, we specify a range of values within which we expect the population parameter to fall. You often encounter such intervals in real life, although they are usually phrased in terms of “plus or minus” some amount (called the margin of error). For example, the evening news may report that a sample survey showed that 45% of the voters support the president, with a mar- gin of error of plus or minus 3%. This means that the pollsters expect that, if they asked the entire population, the result would be within ;3% of 45%: They believe that the true portion of the population that supports the president is inside the interval that is between 42% and 48%. We will perform interval estimation in a similar way by creating a confidence inter- val. Confidence intervals can be used to describe various population parameters, but the most common is for a single. The confidence interval for a single describes a range of values of , one of which our sample mean is likely to represent. For example, intuitively we know that sampling error is unlikely to produce a sample mean of 65. Thus, a sample mean is likely to represent any that the mean is not significantly dif- ferent from. The logic behind a confidence interval is to compute the highest and low- est values of that are not significantly different from the sample mean. All s between these two values are also not significantly different from the sample mean, so the mean is likely to represent one of them. This is because we must be sure that our sample is not representing the described in H0 before we estimate any other that it might represent. Computing the Confidence Interval The t-test forms the basis for the confidence interval, and here’s what’s behind the for- mula for it. We seek the highest and lowest values of that are not significantly differ- ent from the sample mean. The most that can differ from a sample mean and still not be significant is when tobt equals tcrit. We can state this using the formula for the t-test: X 2 tcrit 5 sX To find the largest and smallest values of that do not differ significantly from our sample mean, we determine the values of that we can put into this formula along with our X and sX. Because we are describing the above and below the sample mean, we use the two-tailed value of tcrit. Then by rearranging the above formula, we create the formula for finding the value of to put in the t-test so that the answer equals 2tcrit. We also rearrange this formula to find the value of to put in so that the answer equals 1tcrit. Our sample mean represents a between these two s, so we combine these rearranged formulas to produce: The formula for the confidence interval for a single is 1sX212tcrit2 1 X # # 1sX211tcrit2 1 X The symbol stands for the unknown value represented by the sample mean. Find the two-tailed value of tcrit in the t-tables at your for df 5 N 2 1, where N is the sample N. Returning to our previous diagram, we replace the symbols low and high with the numbers 59. On the other hand, there is a 95% chance 311 2 2110024 that the being represented is within this interval. Therefore, we have created what is called the 95% confidence interval: We are 95% confident that the interval between 59. Notice, however, that greater confidence comes at the cost of less precision: This interval spans a wider range of val- ues than did the 95% interval, so we have less precisely identified the value of. Usu- ally, researchers compromise between precision and confidence by creating the 95% confidence interval. Thus, we conclude our one-sample t-test by saying, with 95% confidence, that our sample of men represents a between 59. In fact, there- fore, you should compute a confidence interval anytime you are describing the repre- sented by the mean of a condition in any significant experiment. With N 5 22, you perform a one-tailed test any one of which our X is likely to represent.  