R2.7 Standard Normal areas Use Table A to find the proportion of observations from a standard Normal distribution that falls in each of the following regions. In each case, sketch a standard Normal curve and shade the area representing the region.

(a) $z\le -2.25$

(b)$z\ge -2.25$

(c) role="math" localid="1649401025226" $z>1.77$

(d) $-2.25<z<1.77$

Use Table A to find the proportion of observations from a standard Normal distribution

$z\le -2.25=?$

Table A contains probabilities of values smaller than a z-score.

Thus the probability is given in table A in the row with $-2.2$and in the column of 5.

$P(z<-2.25)=0.0122$

$P(Z<-2.25)$: in a z-table having area to the left of z, locate $-2.2$ in the left most column. Move across the row to the right under column $0.05$and get value$0.0122$

Use Table A to find the proportion of observations from a standard Normal distribution

$z\ge -2.25=?$

Since this is continuous distribution, $P(Z\ge -2.25)=P(Z>-2.25)$

Area to the right of $-2.25$

$P(7\ge -2.25)=1-P(7<-2.25)$

$P(Z>-2.25)=1-P(Z<-2.25)=1-0.0122=0.9878$

Hence, the value we get$P(Z\ge -2.25)=0.9878$.

Use Table A to find the proportion of observations from a standard Normal distribution

$z>1.77=?$

Since this is continuous distribution, $P(Z\ge 1.77)=P(Z>1.77)$

Area to the right of $1.77$

$P(Z>1.77)=1-P(Z<1.77)$

$P(Z>1.77)=1-P(Z<1.77)=1-0.9616=0.0384$

Hence,$P(Z<1.77)$: in a z-table having an area to the left of z, locate 1.7 in the left most column. Move across the row to the right under column 0.07 and get value 0.9616

Use Table A to find the proportion of observations from a standard Normal distribution

$-2.25<z<1.77=?$

Area in between $-2.25$and $1.77$

$P(-2.25<Z<1.77)=P(Z<1.77)-P(Z<-2.25)$

$P(-2.25<Z<1.77)=0.9616-0.0122=0.9494$

Hence, we get$P(-2.25<Z<1.77)=0.9494$.