Math 160 - Cooley Intro To Statistics OCC
Section 6.3 – Working With Normally Distributed Variables
To determine a Percentage or Probability for a Normally Distributed Variable
Step 1: Sketch the normal curve associated with the variable.
Step 2: Shade the region of interest and mark its delimiting x-value(s).
Step 3: Compute the z-score(s) for the delimiting x-value(s) found in Step 2.
Step 4: Use table II to find the area under the standard normal curve delimited by the z-score(s) found in Step 3.
The 68.26-95.44-99.74 Rule (or Empirical Rule)
Any normally distributed variable has the following properties.
Property 1: 68.26% of all possible observations lie within
one standard deviation to either side of the
mean, that is, between μ – σ and μ + σ.
Property 2: 95.44% of all possible observations lie within
two standard deviations to either side of the
mean, that is, between μ – 2σ and μ + 2σ.
Property 3: 99.74% of all possible observations lie within
three standard deviations to either side of the
mean, that is, between μ – 3σ and μ + 3σ.
To Determine the Observations Corresponding to a Specified Percentage or Probability for a Normally Distributed Variable
Step 1: Sketch the normal curve associated with the variable.
Step 2: Shade the region of interest.
Step 3: Use table II to obtain the z-score(s) for the delimiting x-values found in Step 2.
Step 3: Find the x-value(s) having the z-score(s) found in Step 3.
J Exercises:
Assume that the amount of time children spend watching television per year is normally distributed with a mean of 1600 hours and a standard deviation of 100 hours.
1) What percent (or probability) of children watch television less than 1750 hours per year?
2) What percent (or probability) of children watch television more than 1480 hours per year?
3) What percent (or probability) of children watch television between 1650 and 1750 hours per year?
J Exercises:
Assume that the amount of time to prepare and deliver a pizza from Domino’s Pizza is normally distributed with
a mean of 20 minutes and standard deviation of 5 minutes.
4) Find the percent (or probability) of pizzas that were prepared and delivered in less than 18 minutes.
5) If Domino’s advertises that the pizza is free if it takes more than 30 minutes to deliver, what percent
(or probability) of the pizza will be free?
An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assume that bulb life is normally distributed.
6) 68.26% of all manufactured light bulbs will last between ______and ______days.
7) 95.44% of all manufactured light bulbs will last between ______and ______days.
8) 99.74% of all manufactured light bulbs will last between ______and ______days.
9) What is the probability that an Acme light bulb will last at most 365 days?
10) Obtain the 90th percentile for the life of a light bulb manufactured by the Acme Corporation.
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