For the first question,
Suppose you are managing a portfolio of stocks. You want to see how well your portfolio performs using historical data, and one way to measure such performance is by calculating the average expected return on the portfolio. There are different definitions of average, but this will not matter here. The point is that this calculation will give you one, definite number. You quickly realize, however, that this number might not be a good indicator of performance because of market fluctuations. There is randomness in the data which you can’t avoid by taking a simple average. Instead, you want to get an interval - a confidence interval - for which you are pretty confident it contains the true value of the mean. Usually, finding a confidence interval is difficult, but in the case of stock returns, people often make the simplifying assumptions that the stock returns are independent and that they follow a normal distribution. This allows you to use the existing knowledge of standard normal distribution to find the confidence interval you need.
Any statistic textbook will give the confidence interval for the mean of a normally distributed data as
sample_mean ± Z _{\alpha/2}* \sigma/\sqrt(n)
where
\sigma is the population standard deviation of the stock return;
Z _{\alpha/2} is the critical Z value, which can be found on a Z-table;
and n is the number of observations (the number of stock returns you observe).
the quantity to the right of the critical value for Z is the “standard error”
that is, standard error = \sigma/\sqrt(n)
as Bill noted above, when n = 1, this reduces to
standard error = \sigma
technically, if you don’t know the population standard deviation, you need to use the unbiased sample standard deviation and a t-distribution with n-1 degrees of freedom, but I don’t think this is common, at least for exam purposes.
Z _{\alpha/2}, the critical Z value, is the inverse of the normal cumulative distribution function
evaluated at 1 - \alpha/2. Translation: It’s the area under the standard normal distribution curve from the left all the way to the right until x = Z _{\alpha/2}. Remember that a standard normal distribution is a bell-shaped curve, symmetric around its mean, has mean of 0 and a variance of 1.
For example, when you are calculating a 90% confidence interval,
p = .90 (to avoid confusion: this is not the p-value),
\alpha = 1 - p = 1 - .90 = .10
1 - \alpha/2 = 1 - .10/2 = 1 - .05 = .95
and Z _{\alpha/2} = inverseN(.95) = 1.64485
The reason you take \alpha/2 and not \alpha is because when you calculate a confidence interval, you slice the standard normal distribution intro three parts - the left tail, your confidence interval, and the right tail. Both tails have each an area of \alpha/2 (they are symmetric around the mean of zero), so that combined the tails have an area of \alpha, and your confidence interval has an area of 1 - \alpha.
Another way of finding Z _{\alpha/2} is by using a Z-table. You need to find the z-value that corresponds to a probability of 95%. You might not actually see the probability of exactly 95% in the Z table, in which case you have to use the average of the two closest values as an approximation.
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The second question pertains to the use of one or two-tailed tests when making a hypothesis test. That is, when is a one or two-tailed test appropriate? There is also confusion on whether to use a significance level of \alpha or \alpha/2.
For any hypothesis test, you always start with a null hypothesis, denoted H_0, where you hypothesize that some quantity you want to test is equal to some given value. You must only use the equal (=) sign for the null hypothesis.
For example,
H_0: \mu = 25.5
The alternative hypothesis is often denoted H_1 and represents a different hypothesis from the null. You must only use the unequal (=/=), less than (<), or greater than (>) sign for the alternative hypothesis.
For example,
H_1: \mu =/= 25.5, or
H_1: \mu < 25.5, or
H_1: \mu > 25.5
A statistics book I’m using makes a fairly simple and easy rule of thumb to remember:
if the alternative hypothesis uses the unequal (=/=) sign, use the two-tailed test;
if the alternative hypothesis uses the less than(<) sign, use the left-tailed test;
finally, if the alternative hypothesis uses the larger than (>) sign, use the right-tailed test
The significance level, \alpha, is the area in the tail of the distribution.
If you are making a left or right-tailed test, the area under the tail is \alpha. If you are making a two-tailed test, the area under BOTH tails is \alpha. Note that because the tails are symmetric, they each have an area of \alpha/2. So for example, when you choose a value of \alpha = .05, you use this value when making a left or right-tailed test, but use .05/2 = .025 when making a two-tailed test.
When \alpha is .05, you use a critical value for Z of inverseN(1 - .05) = inverseN(.95) = 1.64485 (or about 1.645 if you use a Z-table) when making a left or right-tailed test. Because the standard normal distribution is symmetric around the mean of 0, you use -1.64485 for a left-tailed test and +1.64485 for a right-tailed test.
For a two-tailed test, you must use a critical value of
Z = inverseN(1 - .05/2) = inverseN(.975) = 1.95996 (if you use a Z-table, you’ll get a value of 1.96). For a two-tailed test, you must use two critical values - in this case, Z_{left} = -1.95996 and Z_{right} = +1.95996
hopefully this answers your question. 
