Section 8.1 Confidence Intervals for a Population Mean, Standard Deviation Known 351 EXAMPLE 8.2 Construct confidence intervals of various levels A large sample has mean ̄x = 7.1 and standard error ��∕ √ n = 2.3. Construct confidence intervals for the population mean �� with the following levels: a. 90% b. 98% c. 99% Solution The point estimate of �� is ̄x = 7.1. a. From the bottom row of Table A.3, we see that the critical value for a 90% confidence interval is 1.645, so the margin of error is Margin of error = Critical value ⋅ Standard error = (1.645)(2.3) = 3.8 The 90% confidence interval is 7.1 ± 3.8. We can also write this as 7.1 − 3.8 < �� < 7.1 + 3.8, or 3.3 < �� < 10.9. b. From the bottom row of Table A.3, we see that the critical value for a 98% confidence interval is 2.326, so the margin of error is Margin of error = Critical value ⋅ Standard error = (2.326)(2.3) = 5.3 The 98% confidence interval is 7.1 ± 5.3. We can also write this as 7.1 − 5.3 < �� < 7.1 + 5.3, or 1.8 < �� < 12.4. c. From the bottom row of Table A.3, we see that the critical value for a 99% confidence interval is 2.576, so the margin of error is Margin of error = Critical value ⋅ Standard error = (2.576)(2.3) = 5.9 The 99% confidence interval is 7.1 ± 5.9. We can also write this as 7.1 − 5.9 < �� < 7.1 + 5.9, or 1.2 < �� < 13.0. The notation z�� Sometimes we may need to find a critical value for a confidence level not given in the last row of Table A.3. To do this, it is useful to learn a notation for a z-score with a given area to its right. DEFINITION Let �� be any number between 0 and 1; in other words, 0 < �� < 1. ∙ The notation z�� refers to the z-score with an area of �� to its right. ∙ The notation z��∕2 refers to the z-score with an area of ��∕2 to its right. To find the critical value for a confidence interval with a given level, let 1 − �� be the confidence level expressed as a decimal. The critical value is then z��∕2, because the area under the standard normal curve between −z��∕2 and z��∕2 is 1 − ��. See Figure 8.4. zα/2 Area = 1 − α Area = α/2 Area = α/2 −zα/2 Figure 8.4 EXAMPLE 8.3 Find a critical value Find the critical value z��∕2 for a 92% confidence interval. Solution The level is 92%, so we have 1 − �� = 0.92. It follows that �� = 1 − 0.92 = 0.08, so ��∕2 = 0.04. The critical value is z0.04. We now must find the value of z0.04. To do this using Table A.2, we find the area to the left of z0.04. Since the area to the right of z0.04 is 0.04, the area to the left is 1 − 0.04 = 0.96. See Figure 8.5. Area = 0.04 Area = 0.04 Area = 0.92 −z0.04 z0.04 The area to the left of z0.04 is 0.96. Figure 8.5 The critical value z0.04 contains an area of 0.92 under the standard normal curve between −z0.04 and z0.04.
navidi_monk_elementary_statistics_2e_ch7-9
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