352 Chapter 8 Confidence Intervals We now look in the body of Table A.2 (a portion of which is shown in Figure 8.6) to find the closest value to 0.96. This value is 0.9599, and it corresponds to a z-score of 1.75. Therefore, the critical value is z0.04 = 1.75. z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015 1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177 1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319 1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441 1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545 1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633 1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706 1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767 2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817 2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857 2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ Figure 8.6 Figure 8.7 As an alternative to Table A.2, we can find z0.04 with technology, for example, by using the TI-84 Plus calculator. Figure 8.5 shows that the area to the left of z0.04 is 0.96. Therefore, we can find z0.04 with the command invNorm(.96,0,1). See Figure 8.7. Check Your Understanding 1. Find the critical value z��∕2 to construct a confidence interval with level a. 90% 1.645 b. 98% 2.326 c. 99.5% 2.81 d. 80% 1.28 2. Find the levels of the confidence intervals that have the following critical values. a. z��∕2 = 1.96 95% b. z��∕2 = 2.17 97% c. z��∕2 = 1.28 80% d. z��∕2 = 3.28 99.9% 3. Find the margin of error given the standard error and the confidence level. a. Standard error = 1.2, confidence level 95% 2.352 b. Standard error = 0.4, confidence level 99% 1.030 c. Standard error = 3.5, confidence level 90% 5.758 Tech: 5.757 d. Standard error = 2.75, confidence level 98% 6.397 Answers are on page 366. Assumptions The method we have described for constructing a confidence interval requires us to assume that the population standard deviation �� is known. In practice, it is more common that �� is not known. The advantage of first learning the method that assumes �� known is that it allows us to use the familiar normal distribution, so we can focus on the basic ideas of confidence intervals. We will learn how to construct confidence intervals when �� is unknown in Section 8.2. Following are the assumptions for the method we describe. Explain It Again x̄ must be approximately normal: We need assumption 2 to be sure that the sampling distribution of x̄ is approximately normal. This allows us to use z��∕2 as the critical value. Assumptions for Constructing a Confidence Interval for �� When �� Is Known 1. We have a simple random sample. 2. The sample size is large (n > 30), or the population is approximately normal. When these assumptions are met, we can use the following steps to construct a confidence interval for �� when �� is known.
navidi_monk_elementary_statistics_2e_ch7-9
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