5.2The Standard Normal Distribution

5.2The Standard Normal Distribution

Normal Distribution: a distribution of a continuous random variable with a graph that is symmetric and bell-shaped.

Uniform Distribution: a probability distribution in which every value of the random variable is equally likely. The values spread evenly over the range of possibilities. The graph of a uniform distribution results in a rectangular shape. *** Read Class Length Example on page 227.

Density Curve (or probability density function) is a graph of a continuous probability distribution. It must satisfy the following properties:

1. Total area under the curve is 100% or 1 (there is a correspondence

between area and probability).

2. Every point on the curve must have a vertical height that is zero or greater.

*** Read Class Length Example on page 228.

Uniform Density Curves

*** Do #2, pg. 237 *** Do #4, pg. 237

Normal Density Curves

The shapes of these curves are determined by μ and σ.

*** See figure 5-4 on pg. 229.

Standard Normal Distributions

The standard normal distribution is a normal probability distribution with mean μ = 0 and standard deviation σ = 1, and the total area under its density curve is equal to 1.

Finding Probabilities When Given z-Scores

TABLE A-2 gives the area under the standard normal curve to the left of a given z-score. One page of the table uses negative z-scores and the other page uses positives. Each value in the body of the table is a cumulative area from the left up to a vertical line at a specific z-score.

*** Read examples on pages 232 and 233

*** Find the following probabilities. In each case, sketch the graph and shade the corresponding area.

1) P( z < 1) 2) P(z > 2.1)

3) P(z > -.4) 4) P(z < -.29)

5) P(-1.2 < z < 0) 6)P(-1.15 < z < .37)

7) P(1.50 < z <2.5) 8)P(z > 0)

Recall the Empirical Rule from Chapter 2

If the distribution is approximately bell shaped, then:

About ...... of the scores fall within 1 standard deviation from the mean

About ...... of the scores fall within 2 standard deviation from the mean

About ...... of the scores fall within 3 standard deviation from the mean

Now find the following probabilities:

(a) P(-1 < z < 1) (b) P(- 2 < z < 2)(c) P(-3 < z < 3)

What percent of scores is within 1.96 standard deviations of the mean?

Note: If Random Variable is Continuous P( z = a) = 0, for any number a.

Note difference from discrete!!

Note: z scores are positive or negative, but probabilities are from 0 to 1

Finding z Scores from Known Areas (Probabilities)

Often probabilities are given as percents.

***#38, pg. 239

***#40, pg. 239

***#43, pg. 239

5.3 Applications of Normal Distributions

These are more realistic cases where μ ≠ 0 and σ≠ 0.

Finding probabilities (areas)

Step 1: Draw the graph shading the area desired. Label the mean

and the specific x-values being considered.

Step 2: Find the z-Score for each x-value involved.

Z-SCORE:

Step 3: Use table A-2 to find the cumulative left area bounded by z.

Step 4: Answer the problem.

*** #2, pg. 246 IQ Scores

***#4, pg. 246IQ Scores

***#10a, pg. 246 Lengths of Pregnancies

***#18, pg. 248

Finding Values from Know Areas (Probabilities)

Working BACKWARDS

Step 1: Draw the graph, shade and label the given area, and identify the

x-value being sought.

Step 2: Find the cumulative left area bounded by x.

Step 3: Use table A-2 to find the z-score.

(Go backwards! From the main body of the table to the z-score)

Step 4 Find the score, x, by using the formula

*** Do #6, pg. 246 IQ Scores

** *Do #8, pg. 246 IQ Scores

***Do #10b, pg. 247

*** Do #20, pg. 248

Using the TI-83 to obtain probabilities, percentages, areas

Press 2nd, VARS

Select 2:normalcdf(

Typeleft endpoint, right endpoint, , )

Using the TI-83 to obtain normal scores, percentiles

Press 2nd, VARS

Select3:invNorm(

Typetotal area to the left of the desired value, , )

5.4 Which Statistics Make Good Estimators of Parameters?

When using a sample statistic to estimate a population parameter, some statistics are good in the sense hat they target the population parameter and are therefore likely to yield good results. Such statistics are called unbiased estimators. Other statistics are not so good (because they are biasedestimators).

Here is a summary:

Statistics that target population parameters: Mean, Variance, Proportion.
Statistics that do not target population parameters: Median, Range, Standard Deviation. Although the sample standard deviation does not target the population standard deviation, the bias is relatively small in large samples, so s is often used to estimate σ. Consequently, means, proportions, variances, and standard deviation will all be considered as major topics in following chapters, but the median and range will rarely be used.

5.5The Central Limit Theorem

Sampling Distribution of the Mean: The probability distribution of sample means, for a sample of size n.

Central Limit Theorem

Given:

1. That random variable x has a distribution (which may or may not be

normal) with mean μ and standard deviation σ.

2. A simple random sample of size n is selected from this population

(the samples are selected so that all possible samples of size n have the same chance of being selected); then:

Conclusions:

1. The distribution of sample means will, as the sample size

increases, approach a normal distribution.

2. The mean of the sample means is the population mean, μ.

(That is, the normal distribution from Conclusion 1 has mean μ.)

3. The standard deviation of the sample means is

(That is, the normal distribution from Conclusion 1

has standard deviation )

is often called the standard error of the mean.

Notation:N(μ , )

Normal distribution with parameters μ and

Practical Rules:

1. For samples of size n larger than 30, the distribution of the sample

means can be approximated reasonably well by a normal

distribution. The approximation gets better as the sample size n

gets larger.

2. If the original population is itself normally distributed, then the sample

means will be normally distributed for any n (not just those

larger than 30).

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 some sample (or group) be sure to use the value for the standard deviation of the sample means.

Use

*** Read Example pages 262-264

*** Do # 2, pg. 267

*** Do # 8, pg. 268

Interpreting Results

Recall the Rare Event Rule:

If, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct.

*** Read Example pages 265-266

*** Do # 12, pg. 268

*** Do # 14, pg. 269

*** Do # 20, pg. 270

5.6 Normal Approximation to Binomial Distribution

Let’s recall the conditions required for a Binomial Distribution (from section 4.3)

1. The procedure must have a fixed number of trials.

2. The trials must be independent.

3. Each trial must have all outcomes classified into two categories.

4. The probabilities must remain constant for each trial.

Methods used in 4.3

Formula

Table A-1

Calculator: built in program binompdf(n,p,x) or binomcdf(n,p,x)

*** Example: 74% of U.S. households have answering machines. If we select 2500 households at random, what is the probability that more than 650 have answering machines.

P(x > 650) = P(651)+P(652)+P(653)+...+P(2499)+P(2500)

The TI-83 plus may handle this calculation but the regular TI-83 can't.

In this section we are going to use the normal distribution to approximate a binomial distribution.

Continuity Correction Factor

Notice that we are using a CONTINUOUS model to approximate a DISCRETE model. We need to make an adjustment for continuity.

*** Read procedure for continuity corrections on pages 275-6.

*** In the following exercises, the given values are discrete. For each problem do the following 3 steps:

a)Identify the region of the normal curve that corresponds to the indicated probability.

b)Describe the problem in terms of areas, that is,

Area to the right of …….

Area to the left of ………

Area between ….. and ……

c)Give the continuity correction factor.

*** Do #2, p. 279*** Do #4, p. 279

*** Do #6, p. 279*** Do #8, p. 279

Normal Distribution as Approximation to Binomial Distribution

If , and , then the binomial random variable has a probability distribution that can be approximated by a normal distribution with the mean an standard deviation given as

Procedure for Normal Approximation to Binomial

There are 5 Steps to follow:

Back to the example from the first page: 74% of U.S. households have answering machines. Sample 2500 households. Estimate the probability that more than 650 have answering machines.

1) Test if Normal is Appropriate (, and )

np = .74 X 2500 = 1850

nq = .26 X 2500 = 650

Both are greater than 5

2) Find the mean and the standard deviation

3) Draw a normal distribution centered about μ, and identify the region representing the probability to be found.

4) Find the continuity correction factor

5) Estimate the probability

*** Read Example on pages 273-275

*** Read Example on pages 276-278

Interpreting Results

Using Probabilities to Determine when Results Are Unusual

-Unusually high: x successes among n trials is an unusually high number of successes if P(x or more) is very small (such as 0.05 or less)

-Unusually low: x successes among n trials is an unusually low number of successes if P(x or fewer) is very small (such as 0.05 or less)

*** Do #16, pg. 279

*** Do #18, pg. 279

*** Do #22, pg. 280

*** Do #28, pg. 281