Chapter 6
Chapter 6 Normal Probability Distributions
Section 6-1
Probability Density Function (PDF)- graph of a continuous random variable.
Total area under the curve equals 1.
Every point on the curve has a height greater than or equal to 0.
Uniform Distribution- A continuous random variable with the values spread evenly over the range.
Refer to the continuous uniform distribution where wait times are uniformly distributed between 0 and 5 minutes. Assume that a passenger is randomly selected, and find the probability that the wait time is within the given range.
6-1#3 Greater than 3 minutes
Standard Normal Distribution- Bell shaped distribution with mean=0 and standard deviation=1.
Total area under the normal distribution is 1!!
NOTE: THE NORMAL TABLE (TABLE A-2) GIVES AREA/PROBABILITY TO THE LEFT!!!!! These probabilities can also be found using Statcrunch but I recommend using the table first to get the general understanding before using statcrunch.
Video using statcrunch on problems very similar to A through E below(very helpful)
Assume a Standard Normal Distribution and find the following
Video on using Statcrunch on a problem like the below example (helpful!!)
IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.
Find the probability of the following (below these are worked out by using the normal table, but statcrunch can be very helpful here):
Video of using statcrunch for problems like the next 4 examples below (very helpful!)
6-2#6 probability of X>91
IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.
6-2#7 probability of 79 < X < 133
IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.
Find P90
6-2# 31 The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days.
A. In a letter to “Dear Abby”, a wife claimed to have given birth 308 days after a brief visit from her husband, who was working in another country. Find the probability of a pregnancy lasting 308 days or longer. What does the result suggest?
B. If we stipulate that a baby is premature if the duration of pregnancy is in the lowest 3%, find the duration that separates premature babies from those who are not premature. Premature babies often require special care, and this result could be helpful to hospital administrators in planning for that care.
Section 6-4 Central Limit Theorem (See textbook page 267)
To be able to use the CLT, we need one of the following to be satisified:
1) The original distribution has to be normally distributed OR
2) sample size greater than 30
When the Central Limit Theorem requirements are satisfied, the mean remains the same but the standard deviation is reduced by the square root of n... sigma/sqrt(n).
Assume that females have pulse rates that are normally distributed with a mean of 74.0 beats per minute and standard deviation of 12.5 beats per minute.
6-4#5
A) If 1 adult female is randomly selected, find the probability that her pulse rate is less than 80 beats per minute.
B) If 16 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 80 beats per minute. Video on using statcrunch and the Central Limit Thm (little trick we need to use to use for this type problem)
C) Why can the normal distribution be used in part (b), even though the sample size does not exceed 30?
The original distribution was normal (see first part of the problem)