Section 6.1: The Normal Curve

Heights in a group of men are normally distributed with mean μ= 69 inches and standard deviation σ= 3 inches. Find the height whose z-score is 0.6. Interpret the result.

x= 3(.6)+69= 70.8in

Scores on an IQ test are normally distributed with mean μ= 100 and standard deviation σ= 15. What score separates the upper 2% of IQ scores from the lower 98%?

x= invNorm(1-.02, 100, 15)= 130.81

The area under a probability density curve between two values a and b has 2 interpretations:

1. It represents the proportion of a population whose values are between a and b. 2. It can also be interpreted as the probability that a randomly selected value from the population is between a and b.

Find the probability that the waiting time is between 5 and 15 seconds.

A=bh A= 10(1/30)= .33

Find the area to the left of z= 1.26. with calc

Sketch a normal curve, label the point z= 1.26, and shade in the area to the left of it. *ans= .896

normal distribution follows the

empirical rule

Probability density curves comes in many varieties, depending on the characteristics of the populations they represent. Many important statistical procedures can be carried out using only one type of probability density curve, called a

normal curve. ~Normal distributions have 1 mode and the distributions are symmetric around the mode. Normal curves extend infinitely far both to the right and to the left.

Consider a woman whose height is x= 67 inches from a normal population with mean μ= 64 inches and σ= 3 inches. The z-score is

z= x-μ/σ= 67-64/3= 1

Consider the probability density curve with some areas indicated. Find: a) The proportion of the population is between 0 and 1. b) The probability that a randomly selected value will be between 1 and 2. c) The proportion of the population is between 0 and 2. d) The probability that a randomly selected value will be greater than 2.

a) .63 b) .23 c) .86 d) 1-.86= .14

The waiting time at a bus stop for the next bus to arrive is uniformly distributed between 0 and 10 minutes. a) Find the probability that the waiting time is less than 3 minutes. b) Find the probability that the waiting time is greater than 6 minutes. c) Find the probability that the waiting time is between 3 and 8 minutes.

a) area=prob= 3(1/10)= .3 b) A= 4(1/10)= .4 c) A= 5(1/10)= .5

IQ scores are normally distributed a mean of 100 and a standard deviation of 15. a) Find the IQ scores that separate the middle 90% of the scores from the top and bottom 5%.33 b) Find the z-scores that bound the middle 80% of the area under the standard normal curve.

a) x1= invNorm(0.05,100,15)= 75 x2= invNorm(.95,100,15)= 125 b) z= invNorm(.1,0,1)= -1.28 b/c z is symmetric z2= 1.28

a) Find the z-score that has an area of 0.26 to its left. b) Find the z-score that has an area of 0.68 to its right. c) Find the z-score that has an area of 0.74 to the left.

a) z= invNorm(.26,0,1)= -.64 b) z=invNorm(1-.68,0,1)= -.47 c) z= invnorm(.74,0,1)= .64

It is often necessary to find areas under the normal curve other than those specified by

the empirical rule

1. A normal population has mean μ= 3 and standard deviation σ= 2.6. Find the value that has 80% of the population below it. 2. A normal population has mean μ= 3 and standard deviation σ= 2.6. Find the 80thpercentile of the population. 3. A normal population has mean μ= 53 and standard deviation σ= 34. Find the value that has 35% of the population above it. 4. A normal population has mean μ= 27 and standard deviation σ= 12. Find the values that separate the middle 60% of the population from the rest.

1. x= invNorm(.8,3,2.6)= 5.2 2. same 3. x= invNorm(1-.35,53,34)= 66 4. invNorm(.2,27,12)= 17 invNorm(.8,27,12)= 37

The diagram is a probability density curve for a population. a) What proportion of the population is between 4 and 6? b) If a value is chosen at random from this population, what is the probability that it is not between 4 and 6?

a) .16 b) P(not 4-6)= 1-P(4-6)= 1-.16= .84

a) Find the area to the right of z= - 0.58. b) Find the area to the left of z = 0.25. c) Find the area between z = -1.13 and z = 2.02 *4 decimals

always draw the curve! a) normalcdf(-.58, inf, 0, 1)= .7190 b) normalcdf(-inf, .25,0,1)= .5987 c) normalcdf(-1.13,2.02,0,1)= .8491

Probabilities for the uniform distribution are straightforward. The probability of an event corresponds to the

area of a rectangle

A study reported that the length of pregnancy from conception to birth is approximately normally distributed with mean μ= 272 days and standard deviation σ= 9 days. What proportion of pregnancies last less than 259 days?

normalcdf(-inf,259,272,9)= .0743

The curve used to describe the distribution of a continuous random variable is a called a

probability density curve. It tells what proportion of the data falls within a given interval.

Suppose we want to find the value from a normal distribution that has a given z-score. To do this, we

solve the standardization formula z=x-u/o for x. The value of x that corresponds to a given z-score is x= z+μ x σ

Recall that the z-score of a data value represents the number of

standard deviations that data value is above or below the mean. ~We can convert x to a z-score by using a method known as standardization. ~The z-score of x is z= x-μ/σ

Using Table A.2 to Find Areas:

table A.2 may be used to find the area to the left of a given z-score.

Compact fluorescent bulbs are more energy efficient than incandescent bulbs, but they take longer to reach full brightness. The time that it takes for a compact fluorescent bulb to reach full brightness is normally distributed with mean 29.8 seconds and standard deviation 4.5 seconds. A randomly selected bulb takes 28 seconds to reach full brightness. Find and interpret the z-score for x= 28.

z= 28-29.8/4.5= -.4 .4 std dev below the mean

Body temperatures in a sample of8670 men with mean 98.2x=and sample standard deviation s= 0.6.The distribution of temperature measurements is approximately bell-shaped. Compare the proportion of temperatures from the frequency distribution to those predicted by the Empirical Rule.

~68% between 97.6 and 98.8 ~95% between 97 and 99.4 ~99.7% between 96.4 and 100

Standard normal curve:

~A normal distribution can have any mean and any positive standard deviation. ~The distribution with a mean of 0 and standard deviation of 1 is known as the standard normal distribution. ~The probability density function for the standard normal distribution is called the standard normal curve.

Finding Areas with the TI-84 pluse:

~On the TI-84 Plus calculator, the normalcdf command is used to find areas under a normal curve. ~Four numbers must be used as the input. ~The first entry is the lower bound of the area. ~The second entry is the upper bound of the area. ~The last two entries are the mean and standard deviation. ~This command is accessed by pressing 2nd, Vars. *normalcdf(lower, upper, mean, st dev) - if standard is normal mean is 0 and st dev is 1

Normal Values From Given Areas using the calc:

~The invNorm command on the TI-84 Plus calculator returns the value from the normal population with a given area to its left. ~This command takes three values as its input. ~The first value is the area to the left, the second and third values are the mean and standard deviation, respectively. ~This command is accessed by pressing 2nd, Vars. ~some newer calcs will let you choose left/right/center

Properties of Normal curves:

~The population mean determines the location of the peak ~The population standard deviation measures the spread of the population. ~mean=median=mode

Properties of Probability Density Curves:

~The region above a single point has no width, thus no area. Therefore, if X is a continuous random variable, P(X= a) = 0 for any number a. ~P(a< X< b) =P(a≤ X≤ b) ~For any probability density curve, the area under the entire curve is 1, because this area represents the entire population.

Uniform distribution:

~one in which values in any region are equally likely. The probability density curve for a uniform distribution is a horizontal line. `Imagine that when a traffic light turns red, it stays red for 30 seconds before turning green. If you pull up to the light when it is red, the amount of time you will wait before the light turns green is uniformly distributed between 0 and 30 seconds. *area of a rectangle: A=bh