1 NORMAL PROBABILITY DISTRIBUTIONS 2 Overview 3 Normal Distribution
If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped and it can be described by the equation below we say that it has a normal distribution.
4 The Normal Distribution
The curve is bell-shaped and symmetric.
5 The Standard Normal Distribution 6 Requirements for a Probability Distribution
where x assumes all possible values.
for every individual value of x.
7 Density Curve
A density curve is a graph of a continuous probability distribution. It must satisfy the following properties
The total area under the curve must equal 1.
Every point on the curve must have a vertical height that is 0 or greater. (That is the curve cannot fall below the x-axis.)
8 Relationship Between Area Under the Curve and Probability
Because the total area under a density curve is equal to 1 there is a correspondence between area and probability.
9 Probabilities and a Continuous Probability Distribution
For continuous numerical variables and any particular numbers a and b
10 Uniform Distribution
A continuous random variable has a uniform distribution if its values spread evenly over the range of possibilities. The graph of a uniform distribution results in a rectangular shape.
11 Uniform Distribution
The uniform distribution is symmetric and rectangular.
Suppose that the continuous random variable X has a uniform distribution over the interval from 0 to 5. Find the probability that a randomly selected value of X is
More than 3
Less than 1
Between 2 and 4.
13 Standard Normal Distribution
The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1 and the total area under its density curve is equal to 1.
14 Standard Normal Distribution
The standard normal distribution
A probability of falling in an interval is just the area under the curve.
Find the area under the standard normal distribution to the left of 1.5.
Find the area under the standard normal distribution to the right of -2.
Find the area under the standard normal distribution between -2 and 1.5.
Let z denote a random variable that has a standard normal distribution. Determine each of the following probabilities
19 Calculating Probabilities Given a z Score
Table A-2 is designed only for the standard normal distribution which has a mean of 0 and a standard deviation of 0.
Table A-2 is on two pages with one page for negative z scores and the other page for positive z scores.
Each value in the body of the table is a cumulative area from the left up to a vertical boundary above the specific z score.
20 Calculating Probabilities Given a z Score
When working with a graph avoid confusion between z scores and areas.
z score Distance along the horizontal scale of the standard normal distribution refer to the leftmost column and top row of Table A-2.
Area Region under the curve refer to the values in the body of Table A-2.
The part of the z score denoting hundredths is found across the top row of Table A-2.
Determine the z value that separates
the smallest 10 of all the z values from the others
the largest 5 of all the z values from the others.
22 Calculating a z Score Given a Probability
Draw a bell-shaped curve and identify the region under the curve that corresponds to the given probability. If that region is not a cumulative region from the left work instead with a known region that is a cumulative region from the left.
Using the cumulative area from the left locate the closest probability in the body of Table A-2 and identify the corresponding z score.
23 Applications of Normal Distributions 24 Standardizing Scores
If we convert values to scores using then procedures with all normal distributions are the same as those for the standard normal distribution.
Suppose the heights of adult males in the population have a normal distribution with a mean of 70 inches and a standard deviation of 2.8 inches. An adult male is selected at random what is the probability that his height is
26 Example (continued)
less than 72 inches
more than 64 inches
between 64 inches and 72 inches
27 Converting Values in a Nonstandard Normal Distribution to z Scores
Sketch a normal curve label the mean and the specific z values then shade the region representing the desired probability.
For each relevant value x that is a boundary for the shaded region use the z Score formula to convert that value to the equivalent z score.
Refer to Table A-2 and use the z scores to find the area of the shaded region. This area is the desired probability.
28 z Scores and Area
Dont confuse z scores and areas.
Choose the correct side of the graph.
A z score must be negative whenever it is located in the left half of the normal distribution.
Areas (or probabilities) are positive or zero values but they are never negative.
29 Example (continued)
The heights of adult males in the population have a normal distribution with a mean of 70 inches and a standard deviation of 2.8 inches. Find
the 80th percentile.
the 25th percentile.
30 Finding Values From Known Areas
Sketch a normal distribution curve enter the given probability or percentage in the appropriate region of the graph and identify the x value(s) being sought.
Use Table A-2 to find the z score corresponding to the cumulative left area bounded by x. Refer to the body of Table A-2 to find the closest area then identify the corresponding z score.
31 Finding Values From Known Areas
Using the formula enter the values for and the z score found in Step 2 then solve for x.
Refer to the sketch of the curve to verify that the solution makes sense in the context of the graph and in the context of the problem.
32 Sampling Distributions and Estimators 33 Sampling Distribution of a Statistic
The sampling distribution of a statistic (such as a sample proportion or sample mean) is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population. (The sampling distribution of a statistic is typically represented as a probability distribution in the format of a table probability histogram or formula.)
34 Sampling Distribution of the Mean
The sampling distribution of the mean is the probability distribution of sample means with all samples having the same sample size n. (The sampling distribution of the mean is typically represented as a probability distribution in the format of a table probability histogram or formula.)
Suppose our population consists of the three values 1 2 and 5.
Calculate the mean mean median range variance and standard deviation for the population.
Find all possible samples of 2 values.
Calculate the mean median range variance and standard deviation for each sample.
Calculate the mean of the sample means sample medians sample ranges sample variances and sample standard deviation.
Compare the results of d with the results of a.
36 Example (continued) 37 Example (continued) 38 Sampling Variability
The value of a statistic such as the sample mean depends on the particular values included in the sample and it generally varies from sample to sample. This variability of a statistic is called sampling variability.
39 Sampling Distribution of the Proportion
The sampling distribution of the proportion is the distribution of sample proportions with all samples having the same sample size n taken from the same population.
40 Properties of the Sampling Distribution of the Proportion
Sample proportions tend to target the value of the population proportion.
Under certain conditions the distribution of sample proportions approximates a normal distribution.
41 Biased and Unbiased Estimators
A sample statistic is an unbiased estimator of a population parameter if it targets the population parameter.
A sample statistic is a biased estimator of a population parameter if it does not target the population parameter.
42 Which Statistics Make Good Estimators of Parameters
Statistics that target population parameters Mean Variance Proportion
Statistics that do not target population parameters Median Range Standard Deviation
43 The Central Limit Theorem 44 Example
Suppose the heights of adult males in the population have a normal distribution with a mean of 70 inches and a standard deviation of 2.8 inches. An adult male is selected at random what is the probability that his height is less than 68 inches
45 The Central Limit Theorem and the Sampling Distribution of
The random variable x has a distribution (which may or may not be normal) with mean and standard deviation .
Simple random samples all of the same size n are selected from the population. (The samples are selected so that all possible samples of size n have the same chance of being selected.)
46 The Central Limit Theorem and the Sampling Distribution of
The distribution of sample means will as the sample size increases approach a normal distribution.
The mean of all sample means is the population mean . (That is the normal distribution from Conclusion 1 has mean .)
The standard deviation of all sample means is . (That is the normal distribution from Conclusion 1 has standard deviation .)
47 The Central Limit Theorem and the Sampling Distribution of
Practical Rules Commonly Used
If the original population is not itself normally distributed here is a common guideline For samples of size n greater than 30 the distribution of the sample means can be approximated reasonably well by a normal distribution. (There are exceptions such as populations with very non-normal distributions requiring samples sizes much larger than 30 but such exceptions are relatively rare.) The approximation gets better as the sample size n becomes larger.
If the original population is itself normally distributed then the sample means will be normally distributed for any sample size n (not just the values of n larger than 30).
48 Notation for the Sampling Distribution of
If all possible random samples of size n are selected from a population with mean and standard deviation the mean of the sample means is denoted by soAlso the standard deviation of the samples means is denoted by so is often called the standard error of the mean.
Suppose the heights of adult males in the population have a normal distribution with a mean of 70 inches and a standard deviation of 2.8 inches. If a random sample of ten adult males is selected what is the probability that the sample mean is less than 68 inches
50 The Central Limit Theorem The Bottom Line
As the sample size increases the sampling distribution of sample means approaches a normal distribution.
According to the Energy Information Administration the mean household size in the United States in 1997 was 2.6 people with a standard deviation of 1.5 people. What is the probability that a random sample of 100 households results in a sample mean household size of 2.4 or less
52 Applying The Central Limit Theorem
When working with an individual value from a normally distributed population use the methods of Section 5.3. Use
When working with a mean for same sample (or group) be sure to use the value for the standard deviation of the sample means. Use
53 Interpreting Results
Rare Event RuleIf under a given assumption the probability of a particular observed event is exceptionally small we conclude that the assumption is probably not correct.
54 Correction for a Finite Population
When sampling with replacement and the sample size n is greater than 5 of the finite population size N (that is ) adjust the standard deviation of the sample means by multiplying it by the finite population correction factor
55 Normal as Approximation to the Binomial 56 The Binomial Distribution Recap
A binomial probability distribution results from a procedure that meets all the following requirements
The procedure has a fixed number of trials.
The trials must be independent.
Each trial must have all outcomes classified into two categories.
The probabilities must remain constant for each trial.
57 Binomial Distributions p 0.5 n 3 n 4 n 5 n 6
58 Binomial Distributions p 0.5 n 10 n 20 n 30 n 40
59 Binomial Distributions p 0.3 n 10 n 20 n 30 n 40
60 Normal Distribution as Approximation to Binomial Distribution
If a binomial probability distribution satisfies the requirements and then that binomial probability distribution that can be approximated by a normal distribution with mean and standard deviation and with discrete whole number x adjusted with a continuity correction so that x is represented by the interval from to .
61 Continuity Corrections
When we use the normal distribution (which is a continuous probability distribution) as an approximation to the binomial distribution (which is discrete) a continuity correction is made to a discrete whole number x in the binomial distribution by representing the single x value by the interval from to (that is by adding and subtracting 0.5).
According to Information Please almanac 80 of adult smokers started smoking before they were 18 years old. Suppose 100 smokers 18 years old or older are randomly selected. What is the probability that that
Fewer than 70 of them started smoking before they were 18 years old.
Exactly 80 of them started smoking before they were 18 years old.
63 Assessing Normality 64 Normal Quantile Plot
A normal quantile plot (or normal probability plot) is a graph of the points (x y) where each x value is from the original set of sample data and each y value is the corresponding z score that is a quantile value expected from the standard normal distribution.
65 Procedure for Determining Whether Data Have a Normal Distribution
Histogram Construct a histogram. Reject normality if the histogram departs dramatically from a bell shape.
Outliers Identify outliers. Reject normality if there is more than one outlier present.
Normal quantile plot If the histogram is basically symmetric and there is at most one outlier construct a normal quantile plot. Examine the normal quantile plot using these criteria
If the points do not lie close to a straight line or if the points exhibit some systematic pattern that is not a straight-line pattern then the data appear to come from a population that does not have a normal distribution.
If the pattern of the points is reasonably close to a straight line then the data appear to come from a population that has a normal distribution.
Recall our study of bears the data for the lengths of bears is given in Data Set 6 of Appendix B. Determine whether the requirement of a normal distribution is satisfied. Assume that this requirement is loose in the sense that the population distribution need not be exactly normal but it must be a distribution that is basically symmetric with only one mode.
67 Example (continued)
68 Example (continued)
69 Example (continued)
Using the weights of bears (given in Data Set 6 of Appendix B) determine whether the requirement of a normal distribution is satisfied. Assume that this requirement is loose in the sense that the population distribution need not be exactly normal but it must be a distribution that is basically symmetric with only one mode.
70 Example (continued)
71 Example (continued)
72 Data Transformations
For data sets where the distribution is not normal we can transform the data so that the modified values have a normal distribution. Common transformations include
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