# Normal Distributions

*Worksheet 4*

### Exercises

- GPAs of SUNY Oswego freshman biology majors have
approximately the normal distribution with mean 2.87 and
standard deviation .34.
- In what range do the middle 90% of all freshman
biology majors’ GPAs lie?
- Students are thrown out of school if their GPA
falls below 2.00. What proportion of all freshman
biology majors are thrown out?
- What proportion of freshman biology majors have
GPA above 3.50?

- The length of elephant pregnancies from conception to
birth varies according to a distribution that is
approximately normal with mean 525 days and standard
deviation 32 days.
- What percent of pregnancies last more than 600
days (that’s about 20 months)?
- What percent of pregnancies last between 510 and
540 days (that’s between 17 and 18 months)?
- How short do the shortest 10% of all pregnancies
last?

- Wingspans of adult herons have approximate normal
distribution with mean 125 cm and standard deviation 12
cm.
- What proportion of herons have wingspan more than
140 cm?
- What is the median wingspan?
- What are the first and third quartiles of the
wingspans? (25% of all herons have wingspan less
than the first quartile; 25% have wingspan more
than the third quartile.) Use these to obtain the
IQR for heron wingspans.

- When robins’ eggs are weighed, it turns out that
they vary according to approximately the normal
distribution with mean 17.5 mg and standard deviation 3.5
mg.
- What proportion of robins’ eggs weigh
between 10 mg and 25 mg?
- How large are the largest 2% of all robins’
eggs?

### Solutions

- a) From 2.311 to 3.429; b) 0.0052; c) 0.0322
- a) 0.95% (a little less than 1%); b) 36.16%; c) no more
than 484 days.
- a) 0.1056; b) 125 cm (the mean and median are the same
for a normal distribution because of the symmetry); c)
the
*Z*-scores for the quartiles are -0.67 and 0.67.
That is, quartiles for normal distributions are *always*
0.67 times the standard deviation below (Q1) and above (Q3)
the mean=median. So, for this data, Q1 = 140 - 0.67*12 = 148.0
cm and Q1 = 140 + 0.67*12 = 132.0
cm. Then IQR = 148.0 - 132.0 = 16.0 cm.(For
normal distributions IQR = 1.34*s -- always!)
- a) 0.9676; b) 24.68 mg.