Inferential Statistics & Hypothesis Testing Heibatollah Baghi, and Mastee Badii

Objectives Conduct one sample mean test Using Z statistics Using t statistics

Inferential Statistics Usage Researchers use inferential statistics to address two broad goals: Estimate the value of population parameters Hypothesis testing

Distribution of Coin Tosses If you see 10 heads in a row, is it a fair coin?

Sample & Population Think of any sequence of throws as a sample from all possible throws Think of all possible throws as the entire population. One-Sample Inferential Tests estimate the probability that a sample is representative of the total population (within +/- ~2 standard deviations of the mean, or the middle 95% of the distribution).

Logic of Hypothesis Testing Is the value observed consistent with the expected distribution? On average, 100 coin tosses should lead to 50/50 chance of heads. Some coin tosses will be outliers, giving significantly different results. Are differences significant or merely random variations? Statistics is the art of making sense of distributions

Logic of Hypothesis Testing The further the observed value is from the mean of the expected distribution, the more significant the difference

What about this point?

What about this point?

Is this point part of the distribution?

Probability of Membership in a Distribution Depends on location Mean Variance It is a chance event

One-Sample Tests We set a standard beyond which results would be rare (outside the expected sampling error) We observe a sample and infer information about the population If the observation is outside the standard, we reject the hypothesis that the sample is representative of the population

Random Sampling A simple random sampling procedure is one in which every possible sample of n objects is equally likely to be chosen. The principle of randomness in the selection of the sample members provides some protection against the sample unrepresentative of the population. If the population were repeatedly sampled in this fashion, no particular subgroup would be over represented in the sample.

Sampling Distribution The concept of a sampling distribution, allows us to determine the probability that the particular sample obtained will be unrepresentative. On the basis of sample information, we can make inference about the parent population.

Sampling Distribution Sampling Error. No sample will have the exact same mean and standard deviation as the population Sampling distribution of the mean In research sampling error is often unknown since we do not have the population parameters A distribution of means of several different samples of our population Less widely distributed than the population Usually Normal

Population of IQ scores, 10-year olds µ=100 σ=16 n = 64 Sample 1 Sample 2 Sample 3 Etc Is sample 2 a likely representation of our population?

Distribution of Sample Means The mean of a sampling distribution is identical to mean of raw scores in the population (µ) If the population is Normal, the distribution of sample means is also Normal If the population is not Normal, the distribution of sample means approaches Normal distribution as the size of sample on which it is based gets larger Central Limit Theorem

Standard Error of the Mean The standard deviation of means in a sampling distribution is known as the standard error of the mean. It can be calculated from the standard deviation of observations The larger our sample size, the smaller our standard error

Statistical inference Sample of observations Entire population of observations Random selection Parameter µ=? Statistic Statistical inference

Estimation Procedures Point estimates For example mean of a sample of 25 patients No information regarding probability of accuracy Interval estimates Estimate a range of values that is likely Confidence interval between two limit values The degree of confidence depends on the probability of including the population mean

When Sample size is small … A constant from Student t Distribution that depends on confidence interval and sample size

HYPOTHESIS TESTING Hygiene procedures are effective in preventing cold. State 2 hypotheses: Null: H0 : Hand-washing has no effect on bacteria counts. Alternative: Ha : Hand-washing reduces bacteria. The null hypothesis is assumed true: i.e., the defendant is assumed to be innocent.

TWO TYPES OF ERROR True False Reject H0 error correct decision Fail to Reject H0

Alpha & Beta Errors Decision Ho is True Ho is False Reject H0 α 1-β Fail to Reject H0 1-α β

Two Types of Error in Admission to ICU Correct decisions Patients admitted to ICU who would have failed if otherwise Patients denied admission who do fine in step down unit Errors Patient admitted who does not need to be there Patient denied admission who needs to be there

Two Types of Error Alpha: α Beta: β Probability of Type I Error P (Rejecting Ho when Ho is true) Beta: β Probability of Type II Error P (Failing to reject Ho when Ho is false)

Power & Confidence Level 1- β Probability of rejecting Ho when Ho is false Confidence level 1- α Probability of failing to reject Ho when Ho is true

Steps in Test of Hypothesis Determine the appropriate test Establish the level of significance:α Determine whether to use a one tail or two tail test Calculate the test statistic Determine the degree of freedom Compare computed test statistic against a tabled value

1. Determine Appropriate Test Level of measurement Number of groups being compared Sample size Extent to which assumption for parametric tests have been met Relatively Normal distribution Approximately interval level variable

2. Establish Level of Significance α is a predetermined value The convention α = .05 α = .01

3. Determine Whether to Use One or Two Tailed Test If the alternative hypothesis specifies direction of the test, then one tailed Otherwise, two tailed Most cases

4. Calculating Test Statistics For one sample tests, use Z test statistic if population is Normal,  is known, or if sample size is large For one sample tests, use T static if population distribution is not known or if sample size is small (less than 30)

5. Determine Degrees of Freedom Number of components that are free to vary about a parameter Df = Sample size – Number of parameters estimated Df is n-1 for one sample test of mean

6. Compare the Computed Test Statistic Against a Tabled Value

Example of Testing Statistical Hypotheses About µ When σ is Known (Large Sample Test for Population Mean).

“Does Home Schooling Affect Educational Outcomes?” Research Question “Does Home Schooling Affect Educational Outcomes?”

Statistical Hypotheses Dr. Tate, a researcher at GMU decided to conduct a study to explore this question. He found out that every fourth-grade student attending school in Virginia takes CAT. Scores of CAT are normally distributed with µ = 250 and σ = 50. Home – schooled children are not required to take this test.

Statistical Hypotheses Dr. Tate selects a random sample of 36 home –schooled fourth graders and has each child complete the test. (It would be too expensive and time-consuming to test the entire population of home-schooled fourth-grade students in the sate.) Step 1: Specify Hypotheses H0: µ = 250 Ha: µ > 250 α = 0.05

Calculated Z Select the sample, calculate the necessary sample statistics n=36 σ =50

Critical Z Determine zα  = 0.05 one sided CI of 95% Refer to the Z table and find the corresponding Z score: Z = 1.65

Make Decisions Regarding Ho Because the calculated z is greater than the critical z, Ho is rejected. 1.80 > 1.65 and Ha is accepted The mean of the population of home-school fourth graders is not 250.

Alternative Steps Step 1: Specify Hypotheses Ho: µ = 250 Ha: µ > 250 α = .05 Step 2: Select the sample, calculate sample statistics n=36 σ =50

Using P value to Reject Hypothesis Step 3: Determine the p-value . A z of +1.80 corresponds to a one tailed probability of 0.036. Step 4: Make decision regarding Ho. Because the p-value of 0.036 is less than α =0.05 H0 is rejected. The mean of the population of home-school fourth graders is not 250.

DECISION RULES In terms of z scores: If Zc > Zα Reject H0 In terms of p-value: If p value < α Reject H0

The One-sample Z Test One-Sample tests of significance are used to compare a sample mean to a (hypothesized) population mean and determine how likely it is that the sample came from that population. We will determine the extent to which they occur by chance. We will compare the probability associated with our statistical results (i.e. probability of chance) with a predetermined alpha level.

The One-sample Z Test If the probability is equal to or less than our alpha level, we will reject the null hypothesis and conclude that the difference is not due to chance. If the probability of chance is greater than our alpha level, we will retain the null hypothesis and conclude that difference is due to chance.

Take Home Lesson Procedures for Hypothesis Testing and Use of These Procedures in One Sample Mean Test for Normal Distribution