PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH

Lesson 7Continuous Distributions 

Hints

Suppose that a school band….

90 80 70 60 70 60 50 50 40 30 40 30 2011

100 15 15 15 15 15 15 15 15 15 15 15 15 15

15

One class is not included

Two classes are not included

Three classes are not included Or consisting of only one class

Hints

1

11 1

Pr( ) Pr( )n nn

i i iii i

A A A

These are disjoint

Corrections

Find the probability of the subset of points such that

21y x

Question

Two boys A and B throw a ball at a target. Suppose that the probability that boy A will hit the target on any throw is 1/3 and the probability that boy B will hit the target on any throw is ¼. Suppose also that boy A throws first and the two boys take turns throwing. Determine the probability that the target will be hit for the first time on the third throw of boy A.

Question

If A and B are independent events and Pr(B)<1,

Pr( ) ?c cA B

Question

Suppose that a random variable X has discrete distribution with the following probability function:

Find the value of the constant

2 for 1,2,..

( )0 otherwise

cx

f x x

The probability density function (p.d.f.)

Every p.d.f f must satisfy the following two requirements

Ex: Suppose that X has a binomial with n=2 and p=1/2. Find f(x) and

+

-

( ) 0, for all x,

f(x)dx=1

f x

Pr( 1,5)X

( ) ( )f x P X x

Example

EX: Suppose that the p.d.f of a certain random variable X is as follows:

Find the value of a constant c and sketch the p.d.f.Find the value of

Sketch probability distribution function

2 for 1 2( )

0 otherwise

cx xf x

3Pr( )

2X

Normal p.d.f.

2 2( ) / 2

( ) , 0, ,2

xef x x

Example

EX:Let we have a normal distribution with mean 0 and variance 1.

Find

(0 2)

( 2 2)

(0 1,53)

P X

P X

P X

Example

Adult heights form a normal distribution with a mean of 68 inches and standard deviation of 6 inches.

Find the probability of randomly selecting individual from this population who is taller than 80 inches?

The distribution of sample means

The distribution of sample means is the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population.

A sampling distribution is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.

The distribution of sample means

EX: population: 1, 3, 5, 7

a. Sample size: 2,

b. Pr( 4) ?X

The standard error of

The standard deviation of the distribution of sample means is called the standard error of

1. The standard deviation of the population

2. The sample size

X

X

standard errorX n

Example

A population of scores is normal, with µ=50 and σ=12. Describe the distribution of sample means for samples size n=16 selected from this population

Shape?

Mean?

The distribution of samples will be almost perfectly normal if either one of the following two conditions is satisfied

1. The population from which the samples are selected is normal distribution.

2. The number scores (n) in each sample is relatively large, around 30 or more.

Example

EX: A skewed distribution has µ=60 and σ=8.

a. What is the probability of obtaining a sample mean greater than =62 for a sample of n=4?

b. What is the probability of obtaining a sample mean greater than =62 for a sample of n=64?

X

X

Introduction to hypothesis testing

Hypothesis testing

HP is an inferential procedure that uses sample data to evaluate the credibility of a hypothesis about a population.

Using sample data as the basis for making conclusions about population

GOAL: to limit or control the probability of errors.

Hypothesis testing (Steps)

1. State the hypothesis

H0: predicts that the IV has no effect on the DV for the population

H0: Using constructivist method has no effect on the first graders’ math achievement.

H1:predicts that IV will have an effect on the DV for the population

Hypothesis testing

2. Setting the criteria for a decision

The researcher must determine whether the difference between the sample data and the population is the result of the treatment effect or is simply due to sampling error.

He or she must establish criteria (or cutoffs) that define precisely how much difference must exist between the data and the population to justify a decision that H0 is false.

Hypothesis testing

3. Collecting sample data

4. Evaluating the null hypothesis

The researcher compares the data with the null hypothesis (µ) and makes a decision according to the criteria and cutoffs that were established before.

Decision:

reject the null hypothesis

fail to reject the null hypothesis

X

Errors in hypothesis testing

ACTUAL SITUATION

No effect, H0 True

Effect Exists, H0 False

Researcher decision

Reject H0 Type I error Decision correct

Retain H0 Decision correct

Type II error

Errors

Type I error: consists of rejecting the null hypothesis when H0 is actually true.

Type II error: Researcher fails to reject a null hypothesis that is really false.

Alpha level

Level of significance: is a probability value that defines the very unlikely sample outcomes when the null hypothesis is true.

Whenever an experiment produces very unlikely data, we will reject the null hypothesis.

The Alpha level defines the probability of Type I error.

Critical region

It is composed of extreme sample values that are very unlikely to be obtained if the null hypothesis is true.

.05, 1,96

.01, 2,58

z

z

Significance

A psychologist develops a new inventory to measure depression. Using a very large standardization group of normal individuals, the mean score on this test is µ=55 with σ=12 and the scores are normally distributed. To determine if the test is sensitive in detecting those individuals that are severely depressed, a random sample of patients who are described depressed by a threapist is selected and given the test. Presumably, the higher the score on the inventory is, the more depressed the patient is. The data are as follows: 59, 60, 60, 67, 65, 90, 89, 73, 74, 81, 71, 71, 83, 83, 88, 83, 84, 86, 85, 78, 79. Do patients score significantly different on this test? Test with the .01 level of significance for two tails?