Chapter 6 Statistics

Expected Value Equation: When Using a Uniform Continuous Probability Distribution - Give equation and label each notation

E(x) = (a + b) / 2 Where: a=min value b=max value

In what ways does Bell-Shaped Curve differ from another?

Either the mean μ or the standard deviation σ. could be different or both - The standard deviation determines how flat and wide the normal curve is. Larger values of the standard deviation result in wider, flatter curves, - The mean shifts the bell shape from left to right on the x axis STUDY THE PIC AND SEE how the mean changes it and the standard deviaiton changes the shape

Draw a Bell-Shaped Curve for the Normal Distribution - label both the Y axis and X axis - label the mean with the appropriate notation symbol - draw vertical lines in the approximate location of the standard deviations - label +-standard deviations on the x axis with the appropriate notation symbol - label the % included in each vertical band for the standard deviations 1 through 3

See pic (you should be able to draw this whole thing

Start 6.2

Start 6.2

How do we convert a normal proabiblty distribution (ie. mean d/n = 0) to the standard normal distribution? Use the example: What is the probability that the random variable x is between 10 and 14?

Step 1 - We first assume the mean is equal to the first x (or mean = 10) and compute the z score using the z score equation (see pic). Step 2 -For the second x of 14 we compute its z as z = (14 - 10)/2 = 4/2 = 2 Note: Thus, the answer to our question about the probability of x being between 10 and 14 is given by the equivalent probability that z is between 0 and 2 for the standard normal distribution. Step 3 - Using z = 2.00 and the standard normal probability table inside the front cover of the text, we see that P(z ≤ 2) = .9772. . Because P(z≤ 0) =.5000, we can compute P(.00≤z≤2.00)=P(z≤2)-P(z≤0)= .9772 - .5000 .4772. Hence the probability that x is between 10 and 14 is .4772.

Binomial Experiment with CONTINUOUS Probability Distribution Example: Find the binomial probability of 12 successes in 100 trials assuming a Standard Normal Distribution. With the history of making errors in 10%

Step P(x = 12) for the discrete binomial distribution is approximated by P(11.5 ≤ x ≤ 12.5) for the continuous normal distribution. Then calculate the z scores for the interval assuming the lower is the mean (ie. standard). (SEE PIC FOR CALCULATION) Using the standard normal probability table, we find that the area under the curve to the left of 12.5 is .7967. Similarly, the area under the curve to the left of 11.5 is .6915. Therefore, the area between 11.5 and 12.5 is .7967 - .6915 = .1052. The normal approximation to the probability of 12 successes in 100 trials is .1052. or 10.52% chance that 12 successes will occur.

Variance Equation: When Using a Uniform Continuous Probability Distribution - Give equation and label each notation

Var(x) = ((b - a)^2) / 12 Where: a=min value b=max value

Discrete Random Variable

Variable where the number of outcomes can be counted and each outcome has a measurable and positive probability

Normal Probability Density Function - Give equation and label each notation

Where: μ = mean σ = standard deviation π = 3.14159 e = 2.71828

a) What are the two types of questions that will be asked regarding z score and probabilities for standard normal distributions? b) What is a good tip in answering these type of questions?

a) 1) The question will give you the z score of a value and asks us to use the table to determine the corresponding areas or probabilities. 2) The question provides an area, or probability, and asks us to use the table to determine the corresponding z value. b) Sketching a graph of the standard normal probability distribution and shading the appropriate area will help to visualize the situation and aid in determining the correct answer.

REVIEW Binomial Experiment with Discrete Random Variables a) What is it? b) Give the following equations with correct notation - mean - Standard Deviation

a) An experiment consists of a sequence of n identical independent trials with each trial having two possible outcomes, a success or a failure b) μ = np σ =np(1 - p)

The highest point on the normal curve is at the _____________, which is also the ___________ and ___________ of the distribution.

mean, median, mode

Continuous Random Variable

takes all values in an interval of numbers

standard normal probability distribution - Define - What does the word standard indicate - Why do we use this? - Using the values in the z table what 3 probabilities could we calculate? - But what if our normal probability districution is not standard (ie. mean is not zero)???

- a normal distribution with a mean of zero and a standard deviation of one - That the mean = 0 (note: so without "standard", it would be a normal distribution with a mean other than zero) - By setting the mean equal to zero it makes it makes life a lot easier for us. Probability calculations for this normal distribution have been done and so if you know the z for a x value you just look up the probability that the x will be under or to the left of that z value (See inside cover of book for the table) - (1) the probability that the standard normal random variable z will be less than or equal to a given value (ie. this is the probability straight from the table (2) the probability that z will be between two given values (ie. Subtract the table value from the smaller z from the bigger z) (3) the probability that z will be greater than or equal to a given value (ie. subtract z from 1) - That is, when we have a normal distribution with any mean μ and any standard deviation σ, we answer probability questions about the distribution by first converting to the standard normal distribution.

Z score - define - Equation

- for a particular value "x" the z score is a measure of *how many standard deviations you are away from the mean* - See pic

Chapter 6 deals with the probabilities around CONTINUOUS Random Variables as opposed to Discrete Random Variables. Answer the following: 1) What is the difference between Continuous and Discrete Random Variables? 2) What is different about how probabilities are computed between Discrete and Continuous

1) - Discrete means it is a variable where the number of outcomes can be counted - Continuous means a random variable that may assume any numerical value in an interval so it is limitless 2) - A Discrete probability function f (x) provides the *probability that the random variable assumes a particular value.* - Continuous random variables probability function is the probability density function meaning that it gives a probability the continuous random variable x assumes a value in that interval. So when we compute probabilities for continuous random variables we are computing the probability that the random variable assumes any value within an interval.

probability density function - Define what its used for

A function used to compute probabilities for a continuous random variable. The area under the graph of a probability density function over an interval represents probability.

Binomial Experiment with CONTINUOUS Probability Distribution a) Why do the Binomial Experiment with Discrete Random Variables equations not apply to a CONTINUOUS probability distribution? b) How do we then estimate probabilities for a Binomial Experiment with CONTINUOUS Probability Distribution?

a) Because with a continuous probability distribution, probabilities are computed as areas under the probability density function. As a result, the probability of any single value for the random variable is always zero b) we approximate the binomial probability by computing the area (ie. probaiblity) of an interval containing the value under the corresponding normal curve. We create this interval by adding and subtracting a continuity correction factor to the value we are estimating a probability for. *SO YOU ARE DOING THE SAME THING AS IN 6.2 but instead of being given the interval you are determining the interval with continuity correction factor.*

uniform probability density function a) Give funciton equation b) What is unique about the graph of this function

a) See Pic Where: a=min value b=max value b) the height of the function is the same for each value of x