Statistics and Probability Paper (SPSS)

 PROBABILITY
4.11.3.2 First Homework on Standardization of Variables
(Module 11 Topic 3 HW1)
Date Due:
Student’s Name:
In these exercises, we speak of “natural units” such as Fahrenheight degrees or feet or kilograms and “standardized units” or stds. Where we may
use x to represent the value of a random variable expressed in natural units,
we will use z as the corresponding value expressed in standardized units. The
conversion between the two values is accomplished by the following formulas:
x = µ + z
z = (Xµ)
Perform the following unit conversions.
1. GPAs of SUNY Oswego freshman biology majors have approximately
the normal distribution with mean 2.87 and standard deviation .34.
(a) Convert a GPA of 2.00 into standardized units relative to the
mean. (Convert to a Z-score.)
Answer:
(b) If a student’s GPA is 2.5 std above the mean, what is the student’s
GPA?
Answer:
2. The duration 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.
(a) Convert a duration of 600 days into standardized units relative to
the mean. (Convert to a Z-score.)
Answer:
(b) If the duration of an elephant pregnancy is -1.5 std, then how
many days did the pregnancy last? (Convert from the Z-score
into an X value.)
Answer:
4.11. MODULE 11-12: DISTRIBUTIONS OF RANDOM VARIABLES AND SAMPLE STATISTICS273
3. A forest products company claims that the amount of usable lumber in
its harvested trees averages 172 cubic feet and has a standard deviation
of 12.4 cubic feet. Assume that these amounts have approximately a
normal distribution.
(a) Convert 150 cubic feet into standardized units relative to the
mean. (Convert from X into a Z-score.)
Answer:
(b) Convert a z-score of 1 std above the mean, into cubic feet. Convert
from a Z-score into an X value.
Answer:
4. At two years of age, sardines inhabiting Japanese waters have a length
distribution that is approximately normal with mean 20.20 cm and
standard deviation 0.65 cm. Convert 19.0 cm and 21.0 cm into standardized units relative to the mean.
Answer: 19cm= std.
Answer: 21cm= std.
 PROBABILITY
4.11.2.3 Second Homework on Continuous Random Variables
(Module 11 Topic 2 HW2)
Date Due:
Student’s Name:
Do the following exercises on graph paper and bring it to the classroom
to hand it in. Writeout your verbal responses on the same piece of paper.
1. Graph a continuous unimodal triangular distribution that is symmetric
about the vertical line x=0. Label the coordinates of the x and y
intercepts with numbers so that the total area beneath the distribution
curve will be correct.
2. Sketch a continuous unimodal distribution with a mode at x=0 and
skewed to the right.
3. Sketch a continuous bimodal distribution that is symmetric about the
vertical line x=0.
4. Sketch a continuous trimodal distribution that is symmetric about the
vertical line x=0.
5. Suppose a continuous random variable X has a distribution that is
symmetric about the vertical line x=0. Suppose P(X < 2) = 0.85. (In
other words, suppose that 85% of the population has an X value below
2.) Find P(X < 2). Explain your reasoning and draw a picture to
illustrate your ideas.