Hypothesis testing and confidence intervals

An analyst is conducting a hypothesis test to determine if the mean time spent on investment research is different from 3 hours per day. The test is performed at the 5% level of significance and uses a random sample of 64 portfolio managers, where the mean time spent on research is found to be 2.5 hours. The population standard deviation is 1.5 hours. The 95% confidence interval for the population mean is A. [1.00 < u < 3.50] B. [0.53 < u < 4.46] C. [2.13 < u < 2.87] Answer: C, the book calculates this is 2.5 ± 1.96(.1875) - 2.5 ± .3675 My question is where are they getting the .1875? I calculated the standard of error at 1.1875 but I can’t find anywhere in the book where they relate much less subtract 1. Also, the decision would be to reject the null hypothesis since z statistic is -2.67 (below -1.96) but how does this range (± 1.96) differ from the confidence interval?

The confidence interval equals 2.5 ± (1.96)(1.5/8) The standard error equals sample/population standard deviation (which ever one is given) divided by the square root of n. We reject the null hypothesis because the hypothesized population mean (U = 3) does not lie within the computed confidence interval at the 5% significance level.

The next question states, The analyst should most appropriately A reject the null hypothesis B fail to reject the null hyptothesis C reach no conclusion because the sample standard deviation was not given The answer is A, reject the null hypothesis, however in the answer it stays since -2.67 is less than 1.96, reject Ho. ± 1.96. This is less than the computed confidence interval of ±2.5. So i’m confused because I’m seeing two ranges, ±2.5 which is calculated as (1.96)(1.5/8) and ±1.96 which is just the initial 95% confidence interval which is what they used to determine the null as rejected.

I think you got confused, like I did, in this post. http://www.analystforum.com/phorums/read.php?11,1045414 Basically, there are 2 ways to do it. If doing confidence interval, use the z-value method. If doing hypothesis, use the z-statistic method. Correct me if i’m wrong though.

For this particular question it is pretty obvious the answer is C from elimination because you know when calculating your CI you’re multiplying 1.96 by (1.5/8). For B to work (std/square of sample) would have to be greater than 1 (meaning you sample size would have to basically equal 1 or 2, but it’s 64). For A to work 1.96 multiplied by (1.5/8) would have to equal exactly 1.5 (a very “easy” number).

By no means did I mean to say you should instantly choose C because of that, though. That’s merely how I would initially look at it, and then I would probably do 1.96*(1.5/8) to see if it is a more complex number than “1.5”

When I was doing Hypothesis questions, I found them to a time consuming, especially for a multiple choice exam. I am not emphasizing to much on this part of the section.