Confidence Intervals in Hypothesis Testing

Looking over two problems here in the Hypothesis Testing reading in Schweser and I’m seeing them determine whether or not to reject the null two different ways. In the first example they take the test statistic and multiply it by by the confidence interval (+ and - 1.96) to determine the range which then they use to decide whether to reject or not. In example 2, they simply calculate the z statistic (essentially what is first done in example 1) and then decide to reject it if that value itself is between + or - 1.96. Can someone help explain why or what the logic is here? See below Example 1: Research has gathered data on daily returns of portfolio of call options over a 250 day period. The mean daily return has been .1% and the sample standard deviation of the daily portfolio returns is .25%. The researcher believes the mean daily portfolio return is not equal to 0. Construct a 95% confidence interval for the population mean daily return of the 250-day sample period. Use a z-distribution. Decide if the hypothesis should be rejected. Ho: u is equal to 0 vs Ha: u is not equal to zero. Solution: .25/squareroot of 250 = .0158% standard error .1 - 1.96(.0158) < u < .1 + 1.96(.0158) = .069% < u < .1310 answer: Reject hypothesis Example 2: When a company’s gizmo machine is working properly,the mean length of gizmos is 2.5 inches. From time to to time the machine gets out of alignment and produces gizmos that are either too long or too short. To check the machine, quality control takes a gizmo sample each day. Today a random sample of 49 gizmos showed a mean lengh of 2.49 inches. The population standard deviation is known to be .021 inches. Using a 5% significance level determine if the machine should be shut down or adjusted. Ho: u is equal to 2.5, Ha u is not equal to 2.5 Solution: .021 / square root of 49 = .003 z-statistic, 2.49 - 2.5 / .003 = = -3.33 answer: Reject Hypothesis

I guess example 1 is clear and the question is for example 2. You could do example 2 similar to example 1, if you wanted. Here is how; Your standard error is .021/square root of 49 = .003 So, your interval at 95% confidence is 2.5 - (1.96 x .003) and 2.5 + (1.96 x .003) which is between 2.49412 and 2.50588 Since, today’s sample mean of 2.49 does not fall within this interval, it means, 2.49 does not eastimate population mean of 2.5 at 95% confidence level. So, the machine does need adjustment. The solution which you have posted does the same thing, but in another way. There it is trying to see how far is 2.49 (sample mean) value away from 2.5 (population mean). It is calculating how many standard deviations away, is 2.49 from 2.5. It calculates it as -3.33. We know, if 2.49 was within 1.96 standard deviations from 2.5, it would have been a correct estimate for 2.5 value at 95% confidence. But, since it is further away than that, it is not a true estimate for population mean. Hence, machine needs to be adjusted. It could be a little tricky, but once you have done more questions on this, it will start to fit in correctly.