MFGE 341Quality Science Statistics

What do we remember?

Normal Distributions

So What is a Normal Distribution

• A continuous probability distribution for a random variable ‘x’– Its’ graph is called the normal curve

Normal Curves

Continuous Probability Distributions

• Because a normal distribution is continuous, we cannot use a histogram to graph the data.

• We can use a different type of graph called a probability density function (or pdf for short)– The total area under a pdf curve is equal to 1– The pdf curve can never be negative

The Normal Curve

• The normal curve is a specific pdf curve with the equation

Properties of a Normal Distribution

• The mean, median, and mode are all equal• The normal curve is ‘bell-shaped’ and symmetric about

the mean• The total area under the normal curve equals 1• The normal curve approaches but never reaches the x-

axis• Within 1 standard deviation of the mean, the normal

curve curves downward, outside of 1 standard deviation it curves upward– This means there is an inflection point at 1 standard deviation

Let’s Standardize

• If we standardize our data so the we have a mean of zero and a standard deviation of 1, we get the graph of a standard normal distribution.– We did this before using the z-score

z

Properties of Standard Normal Distributions

• The cumulative area is close to 0 for z-scores close to z=-3.49

• The cumulative area is 0.5 for z=0• The cumulative area is close to 1 for z-scores close to z-3.49

– As the z-scores increase, the cumulative area increases

What if we want to find the area somewhere in-between?

Guidelines

• Draw the picture• Plot the z-score that you are interested in• Shade the area– If shaded left, use the value in the table– If shaded right, use 1-the value in the table– If shaded between two values, subtract the left

value from the right value

What about Probability?

• The probability of a normally distributed event occurring is equal to the appropriate area under the normal curve.

Let’s do the 2-step

• First convert the upper and/or lower bounds to z-scores

• Second, find the area under the normal distribution utilizing those z-scores

Let’s do the 2-step

• You can also go through the process backwards.– If you are given an area or probability, you can find

the corresponding z-score from the table.– Once you have the z-score, you can determine the

desired value from the mean and standard deviation.

Central Limit Theorem

• The central limit theorem describes the relationship between the sampling distribution of sample means and the population mean

Sampling Distribution

Central Limit Theorem

• If the population is normally distributed than the sampling distribution of sample means is normally distributed

• If we do not know how the population is distributed, but our sample size is at least 30, then the sampling distribution of sample means is approximately normal

Central Limit Theorem

• The mean of the sample means is equal to the population mean

• The standard deviation of the sample means is equal to the standard deviation of the population divided by the square root of the sample size

Mean Probability

• To find the probability that a sample mean will lie in a given interval of the sample distribution, convert it to the z-score

Approximating a Binomial Distribution

• What happens when we need to calculate a binomial distribution on a large number of events?– Too much work to use the binomial distribution

formulas– Try to approximate the binomial distribution as a

normal distribution

Can we do that?

• If and , and ‘x’ is a random variable, than we can do a normal approximation– Our mean would be – Our standard deviation would be

Maybe not

• Because you are using a continuous distribution to approximate a discrete distribution, there is some error.

• We need a continuity correction factor– Move .5 units to the left and right• Which direction depends on the case

• Now we can find the probability

5 Easy Steps

• After we recognize that we have a binomial distribution, to find the probability, we:– Determine if we can use a normal approximation– Find the mean and standard deviation– Apply the appropriate continuity correction– Translate to z-scores– Look up the probability in the table