Calculating Inverse PDF of Z term minus a value

Question

I'm finding myself a pick stuck with this, I'm only used to seeing pdf's in terms of $\Phi\left(Z_\frac{\alpha}{2}\right)$, but I've come across the below and am not sure how to calculate.

$\Phi\left(Z_\frac{.05}{2}-3\right)=0.149$

Answer 1

In R statistical software pnorm (without modifying arguments) is the standard normal CDF $\Phi$ and qnorm is the inverse CDF or quantile function. In R your computation is as follows:

q = qnorm(.975);  q
## 1.959964
pnorm(q)
## 0.975
pnorm(q - 3)
## 0.1491616

Notice that the notation $Z_{.025} = 1.96$ involves cutting 2.5% from the upper tail of standard normal. This notation is often used in connection with printed tables.

$P(Z > 1.96) = 0.025,$ whereas $P(Z \le 1.96) = 0.975.$

In the figure below, the area under the density curve to the right of the red dashed line is 0.025 and the area to the left of the purple dotted line is 0.149.

# code for figure
curve(dnorm(x), -3, 3, lwd=2, ylab="PDF", xlab="z", main="Standard Normal Density")
abline(h=0, col="green2")
abline(v = 1.96, col="red", lty="dashed")
abline(v = 1.96-3, col="purple", lty="dotted")