How to I use the standard normal table to get the following Z value?

Question

So I am given that $P(X \le 31.5) = 0.05$ and according to the textbook, after standardizing and using the standard normal table we get

$$(31.5 - \text{mean})/(\text{standard deviation}) = -1.645.$$

Can someone explain to me how they got -1.645 from 0.05? I udnerstand that the left side is just plugging in $X$ into the formula, but how did we turn 0.05 into -1.645?

Answer 1

Using a standard normal table, which you can find a good example of one here: https://www.stat.tamu.edu/~lzhou/stat302/standardnormaltable.pdf,

for $P(x < 31.5) = 0.05$, that means the area to the left of the z-score is $0.05$. Because we know that the distribution is normal, we would go into the middle of the table and find the closest value to $0.05$ as possible. So if we observe $z_1 = -1.64$ and $z_2 = -1.65$, the given areas to the left are $0.05050$ and $0.4947$ respectively.

To get a value as $close$ to $0.05$ as possible, we can interpolate by taking the midpoint of the two values. So $z_m = {1\over2}{(z_1 + z_2)} = -1.645$.

Answer 2

Different books have different formats for normal tables. It will go something like this. The normal table has cut-off points in the margins and probabilities in the body.

Look in the vertical margin for 1.6. Under the column headed .00 you will see .0548. That is the probability corresponding to cutoff 1.60.

But you want the probabilities corresponding to 1.64 and 1.65. So in the same row look under the columns headed .04 and .05 to find numbers .0505 and .0495, respectively.

Finally, the squeeze out the extra decimal place for the the probability corresponding to 1.645, interpolate by averaging (going halfway between) .0505 and .0495 to get .0500. This is a 'lucky' problem because you can get an exact answer. Usually, you just come as close as possible.

Note: Soon you may encounter a t table. Typically, the last row of a t table (perhaps labeled 'infinity') corresponds to the standard normal distribution. If your text has a t table you might look at its last row to see how your question is answered there.

Addendum: Clearly, you are looking at printed normal tables in your Question. However, many statistical calculators and statistical computer programs have procedures for finding normal probabilities and quantiles (cut-offs). For example, the particular values in your question can be obtained in R as follows.

 qnorm(.05); qnorm(.95)  # 'qnorm' finds cutoffs for given tail probability
 ## -1.644854  # probability .05 in left tail
 ##  1.644854  # probability .05 in right tail
 pnorm(-1.645); pnorm(1.645)  # 'pnorm' is the standard normal CDF
 ## 0.04998491 # P(Z < - 1.645)
 ## 0.950015   # P(Z < 1.645)

R is easily available (and free) from r-project.org. It is worth having even if you use it only to look up probability values for normal and other frequently used distributions.