normal distributionstatistics
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?
The only way I can see how to answer this question with pencil and paper only--no tables, no calculators--is to have the quantiles of the standard normal distribution memorized:
$$\begin{align*} \Pr[Z \le 0.8] &\approx 0.842 \\ \Pr[Z \le 0.9] &\approx 1.282 \\ \Pr[Z \le 0.95] &\approx 1.645 \\ \Pr[Z \le 0.975] &\approx 1.96 \\ \Pr[Z \le 0.995] &\approx 2.576. \end{align*}$$ Most statisticians should be familiar with all but the first one. The first one is not commonly encountered.
One can also remember the 68-95-99.7 rule (also called the "empirical rule"): $$\begin{align*} \Pr[-1 \le Z \le 1] &\approx 0.68, \\ \Pr[-2 \le Z \le 2] &\approx 0.95, \\ \Pr[-3 \le Z \le 3] &\approx 0.997.\end{align*}$$
But if none of these memorized values apply, then I don't see how it is reasonable to do the calculation with sufficient precision to be meaningful. If you have a calculator such as a TI-83, you can use the invNorm() and normalcdf() functions.
The first plot should look like this:
Now can you draw the second one?
Best Answer
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 $z_m = {1\over2}{(z_1 + z_2)} = -1.645$.