This module is the complementary teacher's guide for the "Normal Distribution" chapter of the Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.
A fair number of students are familiar with the "bell-shaped" curve. Stress that the normal is a continuous distribution like the uniform and exponential. However, the left and right tails extend indefinitely but come infinitely close to the
-axis. It is not necessary to show the probability distribution function for the normal (it is in the book) because there are normal probability tables and technology available for probability and percentile calculations.
Visualize the data
Draw a picture of the normal graph and explain that it is symmetrical about the mean. The shape of the graph depends on the standard deviation. The smaller the standard deviation, the skinnier and taller the graph. A change in the mean shifts the graph to the right or left. The notation for the normal is
. Draw several normal curves (superimposed upon each other). Have students determine how the means and standard deviations are changing.
The normal distribution notation
The standard normal distribution is of special interest.
Notation:
where
= one z-score (the number of standard deviations a value is to the right or left of the mean). The mean is 0 and the variance (and standard deviation) is 1. Any normal distribution can be standardized to the standard normal by the z-score formula:
.
Do an example showing the standardization. For
and
, the values
and
are each
standard deviation to the right (
) of their respective means. Therefore, they both have a z-score of
.
Do an example using the normal distribution and the standardization.
Several studies have shown that the amount of time people stand in line waiting for a bank teller is normally distributed. Suppose the mean waiting time is 3 minutes and the standard deviation is 1.5 minutes. Let
= the amount of time, in minutes, one person stands in line waiting for a teller. Notation:
Find the probability that one person waits in line for a teller less than 2 minutes. Have students draw the picture and write a probability statement. The picture should have the
-axis.
Probability statement:
. If you use the TI-83/84 series, the function
normalcdf(0, 2, 3, 1.5) in 2nd DISTR.
.
If you use the TI83/84 series of articles, use the function
.
The
normal approximation to the binomial is NOT included in this text. With graphics calculators and computer software, it is easy to draw a binomial graph with a small
and then make
, say, 50. Students will see the graph approach the normal. The normal approximation states that if
follows a binomial distribution with number of trials equal to
and probability of success for any trial equal to
, then by adding
to
, you get a new random variable
(
is either
or
) and
follows a normal distribution
. For the approximation to be a good one, you want
,
, and
.
Assign practice
Assign the
Practice in class to be done in groups.
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
