CHAPTER 6 REVIEW

Normal Probability Distribution A normal probability distribution is a

distribution of a continuous random variable

6.1

1. Bell-shaped 2. Symmetric about the

mean and the high point occurs over the mean

3. Most of the area occurs within 3 standard deviations of the mean

4. The area under the curve is 1

5. determines the extent of the spread

Figure 6.1

A Normal Curve

Normal curves with the same mean but different standard deviations

Empirical Rule

The empirical rule describes the area of a normal distribution Approximately 68% of the data lie within the interval Approximately 95% of the data lie within the interval Approximately 99.7% of the data lie within the interval

6.1

Z – Scores

A z-score measures the number of standard deviations a raw scores lies from the mean.

6.2

The Standard Normal Distribution The Standard Normal Distribution has

and

6.2

Area & Probabilities

Table 5 and Table A give areas under a standard normal distribution that are to the left of a specified z – value

After raw scores have been converted to z – scores, the standard normal distribution table can be used to find probabilities associated with intervals of x-values from any normal distribution

To use the calculator: Normalcdf(lower bound, upper bound)

Normalcdf(lower bound, upper bound, )

6.2 and 6.3

Three Cases1. Left Tail Case

2. Right Tail Case

3. Center Case

Area & Probabilities6.2 and 6.3

Normalcdf

Normalcdf

Normalcdf

Inverse Normal Distribution

The inverse normal distribution is used to find z- values given an area/probability.

1. Draw a sketch and find the area to the left based on which case you have

2. Use the table or the calculator Table: Find the area in the body of the table

and list the corresponding z-score (you may need to convert to an x value)

Calculator:

6.3

A1 – A

Sampling Distributions

Sampling distributions give us the basis for inferential statistics. A sampling distribution is a probability

distribution of a sample statistic based on all possible simple random samples of the same size from the same population. There are different types of sampling

distributions Sampling distributions for the sample mean Sampling distribution for the sample proportion

6.4

The Distribution andThe Central Limit Theorem

For random samples of size n, the distribution is the sampling distribution for the sample mean of an distribution with population mean and standard deviation . If the distribution is normal, then the

corresponding distribution is normal. By the central limit theorem, when n is sufficiently

large , the distribution is approximately normal, even if the original distribution is not normal

For both cases:

6.5

Binomial Distribution Approximated by the Normal Distribution

The binomial distribution can be approximated by a normal distribution with and provided that and and a continuity correction is made. Continuity correction1. If is a left point of an interval, subtract 0.5

to obtain the corresponding normal variable .

2. If is a right point of an interval, add 0.5 to obtain the corresponding normal variable .

6.6

distribution

For binomial trials with probability of success on each trial, the distribution is the sampling distribution of the sample proportion of successes. When and , the distribution is

approximately normal with and

6.6

Assignment

Page 319 #1, 2, 4, 6, 7, 8, 10, 11 – 25 odd