H2 Maths 6.3 - Normal Distribution: Finding Probabilities with a GC

Step 1 - Set up the distribution

The weight X kilograms of a bag of rice is modelled as X tilde N of five, zero point zero four.

Step 1 - Set up the distribution

So the mean is five and the standard deviation is the square root of zero point zero four, which is zero point two.

Step 1 - Set up the distribution

We will find P of X less than five point three, and P of four point eight less than X less than five point two.

Step 2 - Standardise for P(X < 5.3)

To find the probability, convert the boundary to a Z-score.

Step 2 - Standardise for P(X < 5.3)

Z = (5.3 − 5.0) / 0.2 = 1.5.

Step 2 - Standardise for P(X < 5.3)

So we need P of Z less than one point five from the standard normal.

Step 3 - Read the probability from the GC

On your graphing calculator, use normalcdf with lower limit negative one times ten to the ninety-nine, upper limit one point five, mean zero, standard deviation one.

Step 3 - Read the probability from the GC

The calculator returns zero point nine three three two to four decimal places.

Step 3 - Read the probability from the GC

Equivalently, input the original boundaries directly: lower negative one times ten to the ninety-nine, upper five point three, mean five, standard deviation zero point two - same answer.

Step 4 - Find P(4.8 < X < 5.2)

For the two-sided probability, standardise both limits.

Step 4 - Find P(4.8 < X < 5.2)

Z-scores: (4.8 − 5) / 0.2 = −1 and (5.2 − 5) / 0.2 = 1.

Step 4 - Find P(4.8 < X < 5.2)

By symmetry this equals two times P of Z less than one, minus one, which the calculator gives as zero point six eight two seven.

Step 5 - Using symmetry for tail probabilities

The symmetry of the normal curve gives a useful shortcut for tail probabilities.

Step 5 - Using symmetry for tail probabilities

P of X greater than five point three equals one minus P of X less than five point three, which is one minus zero point nine three three two, giving zero point zero six six eight.

Step 5 - Using symmetry for tail probabilities

By symmetry about the mean of five, P of X greater than five point three equals P of X less than four point seven.