Transcript Hints
PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH Lesson 7 Continuous Distributions Hints Suppose that a school band…. 1 1 90 80 70 60 70 60 50 50 40 30 40 30 20 100 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 n Pr( i 1 n Ai ) Pr( Ai i 1 n 1 Ai ) i 1 These are disjoint Corrections Find the probability of the subset of points such that y 1 x2 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( Ac B c ) ? Question Suppose that a random variable X has discrete distribution with the following probability function: c 2 for x 1, 2,.. f ( x) x otherwise 0 Find the value of the constant The probability density function (p.d.f.) Every p.d.f f must satisfy the following requirements f ( x) P( X x) two f ( x) 0, for all x, + f(x)dx=1 - Ex: Suppose that X has a binomial with n=2 and p=1/2. Find f(x) and Pr( X 1,5) Example EX: Suppose that the p.d.f of a certain random variable X is as follows: cx 2 for 1 x 2 f ( x) 0 otherwise Find the value of a constant c and sketch the p.d.f. Find the value of Pr( X 3 ) 2 Sketch probability distribution function Normal p.d.f. f ( x) e ( x )2 / 2 2 2 , 0, , x Example EX:Let we have a normal distribution with mean 0 and variance 1. Find P(0 X 2) P(2 X 2) P(0 X 1,53) 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( X 4) ? The standard error of X The standard deviation of the distribution of sample means is called the standard error of X 1. 2. The standard deviation of the population The sample size standard error X 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 X =62 for a sample of n=4? b. What is the probability of obtaining a sample mean greater than X =62 for a sample of n=64? 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 Collecting sample data 4. Evaluating the null hypothesis The researcher compares the data X 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 3. Errors in hypothesis testing ACTUAL SITUATION Researcher decision No effect, H0 True Effect Exists, H0 False 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, .01, z 1,96 z 2,58 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?