Ch. 12 statistical applications

point estimate

The specific value taken as the estimate of a parameter

effect size

difference between two population means divided by the standard deviation of either population

confidence limit

endpoint of the interval

Central Limit Theorem

factual statement that specifies the nature of the sampling distribution of the mean

Research hypothesis

hypothesis that the study is designed to test, denoted H1

experimental hypothesis

idea that the study was designed to investigate

p-value

probability that a particular results would occur by chance if H0 is true

Interval estimate

range of values approximated to include the parameter

confidence interval

range with a specified probability of including parameter being estimated

Student's t-distribution

sampling generalized function of the t statistic

Central t distribution

sampling generalized function of the t statistic when the thanks hypothesis is true

sampling distribution

the distribution of a statistic over repeated sampling

null hypothesis

the hypothesis to be tested by a statistical test, denoted H0

SAMPLING DISTRIBUTION OF THE MEAN

• Because one sample t tests focus on the sample means, we need to know something about the sampling distribution of the mean : the Central Limit Theorem Given a population with mean and the variance, the sampling distribution of the mean will have a mean equal to u and a variance equal to. The shape of the distribution will approach the normal distribution as N , the sample size , increases . - If you recall from Chapter 8, we have actually learned about it, but we didn't give it a formal name then. If you don't remember it , you can go back to Chapter 8 Lecture Slides ( Slide 9-10 )

HYPOTHESIS TESTING ABOUT MEANS WHEN standard deviation IS KNOWN

• First, we set up our hypotheses : HO : Kids with depressed parents have same levels of depression / anxiety , from normal kids . H1 : Kids with depressed parents have different levels of depression / anxiety as normal kids . or HO H1 : . Where represents the mean of the population from which these kids in our sample were actually drawn .

BUT BEFORE WE MOVE ON ... YOU NEED TO RECALL THESE

• Sampling distribution : the distribution of a statistic ( e.g. , the mean ) over repeated sampling . • Standard error : the standard deviation of the sampling distribution of a statistic .

overview pt. 2

( eg . , whether KSU psyc majors ' IQ are different from the national average IQ ) . When we compute z - obtained , the population variance ( hence standard deviation ) is known to us ( eg . we know that the population SD in IQ is 15 ) . ... - We are going review that again , because it leads to one sample t tests , which apply to the case where population variance is unknown . So the good news is ... what you will learn in this chapter is really close to what you have learned and practiced ! You should feel pretty comfortable about it .

OVERVIEW

- There are different kinds of t tests suitable for different situations , and we will talk about three of them . - In this chapter, we will focus on what statisticians call one sample . t test With this type of t test, we want to figure out what we can expect the sample means to look like if we draw many samples from one population . When we introduced hypothesis testing in Chapter 8 , we have already learned to use z - obtained to test whether our sample mean is different from the population mean

HYPOTHESIS TESTING ABOUT MEANS WHEN g IS KNOWN

. Second , we will calculate the statistics , !!!!! This is where many of you got wrong in Project where you put the population SD ( ) as the + computing the - z score for a single raw score , deviation of the population Q. This is different from the standard error ( SE = ) , not the standard 2 when computing z - obtained . The denominator is denominator ,

CENTRAL LIMIT THEOREM : VISUALIZATION :

Normal Distribution with mu=50 and=7.65 Let's say we're interested in kids ' behavioral problems. We know that it is normally + distributed with a mean of 50 and a SD of 7.65 . This is what the normal distribution looks like for the population ( e . , all kids ) .

CENTRAL LIMIT THEOREM pt. 2

According to Central Limit Theorem , as N increases , the shape of the sampling distribution approaches a normal distribution, regardless of the shape of the parent population. - Therefore, if we draw 1000 students each time, the sampling distribution is closer to a normal distribution than if we just draw 100 students each time, even if the distribution of all KSU students age is not normally distributed

degrees of freedom

an adjusted value of the sample size, often N - 1 or N - 2

sampling distribution example/standard error example

For example , we draw a sample of 100 students from all KSU student , and we do this for 10,000 times . So we will have 10,000 samples ( each has 100 students ) , and for each sample we calculate the average age of students in that sample . As a result , we will have a sampling distribution of 10,000 means . ( Each mean is a sample mean , and it deviates more or less from the population mean μ ) • In the example above , we have a sampling distribution of 10,000 means . The standard deviation of this sampling distribution of the mean is the standard error .

CENTRAL LIMIT THEOREM

Let's say the average age of all KSU students (our population) is 23, and the SD is 4.5 . We draw a sample of 100 students from all KSU students and repeat it 5.000 tirpes . So each sample has 100 students (N = 100), and each sample has a sample mean . Drawing 5,000 samples will result in 5,000 sample means. According to Central - Limit Theorem, the mean of these sample means is 23 , and the standard deviation (standard error) is = In other words , the mean of the sampling distribution is 23 , and the SD is 0.45

HYPOTHESIS TESTING ABOUT MEANS WHEN Standard deviation IS KNOWN

Note : we have covered this section in Chapter 8. This is exactly what you did with those worksheets and the second question of Project 2. We will go over it one more time, and you will see how it leads us to one - sample t test . Example : Suppose we know that in the normal children population , the mean for depression / anxiety is 50 (), and the SD is 10 () . We gathered 166 children who have one * de pressed parent . In this sample, their mean depression/anxiety is 55.71 () and the SD is 7.35 (s) . We want to test whether these children with depressed parents are different from the normal children with regard to depression / anxiety . In other words , we want to test whether these children come from a normal population with a mean of 50 and a SD of 10.

CENTRAL LIMIT THEOREM : VISUALIZATION pt. 2

Now we 10,000 samples of 5 kids ( N = 5 ) from the population, calculate a sample mean for each of the 10,000 samples ; and plot these 10,000 resulting sample means . The sampling distribution of the mean will look like this , It is nearly normal . The mean of the distribution is very close to 50 The standard error

CENTRAL LIMIT THEOREM : VISUALIZATION pt. 3

Suppose we repeat the entire procedure, but instead of 5 kids, we draw 30 kids ( N = 30 ) each time. The sampling distribution of the mean becomes like this : The distribution is again nearly normal . The mean of the sampling distribution is very close to 50 again .. However, the standard error has decreased to 1.39 ( 7.56 / = 1.39 ) . So as N in créases , the standard error decreases .

WHEN DO WE USE ONE SAMPLE T TEST ?

When we want to figure out whether our sample mean is different from the population mean , for instance Whether rats with their amygdala ( sample ) altered display less fear responses than regular rats ( population ) Whether children with abusive parents ( sample ) display more aggressive behavior than children . in normal environment ( population ) Whether KSU students ( sample ) are younger than American college students ( population )

WHEN DO WE USE ONE SAMPLE T TEST? Pt. 2

You may be thinking : wait a minute , we leamed about computing Z - obtained for this in in Chapter 8 apply to the situation when we know exactly what the population SD is , Chapter 8 and practiced it with Project 2. You are absolutely right ! The examples we used However , in reality , we rarely know population SD , and that calls for one - sample 1 tests .

degree of freedom (df)

the number of independent pieces of information remaining after estimating one or more parameters

standard error

the standard deviation of a sampling distribution of. Statistic