Stats 95 t-Tests Single Sample Paired Samples Independent Samples

t Distributions Same Three Assumptions t dist. are used when we know the mean of the population but not the SD of the population from which our sample is drawn t dist. are useful when we have small samples. t dist is flatter and has fatter tails As sample size approaches 30, t looks like z (normal) dist. Same Three Assumptions Dependent Variable is scale Random selection Normal Distribution

Fat Tails Lose Weight With Larger Sample Size

The Robust Nature of the t Statistics Unfortunately, we very seldom know the if the population is normal because usually all the information we have about a population is in our study, a sample of 10-20. Fortunately, 1) distributions in social sciences often approximate a normal curve, and 2) according to Central Limit Theorem the sample mean you have gathered is part of a normal distribution of sample means, and 3) in practice t tests statisticians have found the test is accurate even with populations far from normal

The Robust Nature of the t Statistics The only situation in which using a t test is likely to give a seriously distorted result is when you are using a one-tailed test and the population is highly skewed.

z Statistic Versus t Statistic When you know the Mean and Standard deviation of a population. E.g., a farmer picks 200,000 apples, the mean weight is 112 grams, the SD is 12grams. Calculate the Standard Error of the sample mean When you do not know the Mean and Standard Deviation of the population E.g., a farmer picks 30 out of his 200,000 apples, and finds the sample has a Mean of 112 grams. Calculate the Estimate of the Standard Error of the sample mean

Scenarios When you would use a Single Sample t test A newspaper article reported that the typical American family spent an average of $81 for Halloween candy and costumes last year. A sample of N = 16 families this year reported spending a mean of M = $85, with s = $20. What statistical test would we use to determine whether these data indicate a significant change in holiday spending? Many companies that manufacture lightbulbs advertise their 60-watt bulbs as having an average life of 1000 hours. A cynical consumer bought 30 bulbs and burned them until they failed. He found that they burned for an average of M = 1233, with a standard deviation of s = 232.06. What statistical test would this consumer use to determine whether the average burn time of lightbulbs differs significantly from that advertised?

Difference Between Calculating z Statistic and t Statistic Standard Deviation of a Sample: Estimates the Population Standard Deviation Standard Error of a Sample: estimates the Sample Error of the Population t Statistic for Single-Sample t Test

Estimating Population from a Sample Main difference between t Tests and z score: use the standard deviation of the sample to estimate the standard deviation of the population. How? Subtract 1 from sample size! (called degrees of freedom) Use degrees of freedom (df) in the t distribution chart Standard Deviation of a Sample: Estimates the Population Standard Deviation

t Distribution Table

Example of Single Sample t Test The mean emission of all engines of a new design needs to be below 20ppm if the design is to meet new emission requirements. Ten engines are manufactured for testing purposes, and the emission level of each is determined. Data: 15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9, 12.7, 13.9 Does the data supply sufficient evidence to conclude that type of engine meets the new standard, assuming we are willing to risk a Type I error (false alarm, reject the Null when it is true)) with a probability = 0.01? Step 1: Assumptions: dependent variable is scale, Randomization, Normal Distribution Step 2: State H0 and H1: H0 Emissions are equal to (or greater than) 20ppm; H1 Emissions are lesser than 20ppm (One-Tailed Test)

Example of Single Sample t Test The mean emission of all engines of a new design needs to be below 20ppm if the design is to meet new emission requirements. Ten engines are manufactured for testing purposes, and the emission level of each is determined. Data: 15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9, 12.7, 13.9 Step 3: Determine Characteristics of Sample Mean = Standard Deviation of Sample = Standard Error of Sample = Step 4: Determine Cutoff df = N-1 = 10-1 =9 t statistic cut-off = -2.822

Example of Single Sample t Test The mean emission of all engines of a new design needs to be below 20ppm if the design is to meet new emission requirements. Ten engines are manufactured for testing purposes, and the emission level of each is determined. Data: 15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9, 12.7, 13.9 Step 3: Determine Characteristics of Sample Mean M = 17.17 Standard Deviation of Sample s = 2.98 Standard Error of Sample sm = 0.942 Step 4: Determine Cutoff df = N-1 = 10-1 =9 t statistic cut-off = -2.822 Step 5: Calculate t Statistic

Example of Single Sample t Test The mean emission of all engines of a new design needs to be below 20ppm if the design is to meet new emission requirements. Ten engines are manufactured for testing purposes, and the emission level of each is determined. Data: 15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9, 12.7, 13.9 Mean M = 17.17 Standard Deviation of Sample s = 2.98 Standard Error of Sample sm = 0.942 Step 5: Calculate t Statistic Step 6: Decide (Draw It) t statistic cut-off = -2.822 t statistic = -3.00 Decide to reject the Null Hypothesis

Paired Sample t Test The paired samples test is a kind of research called repeated measures test (aka, within-subjects design), commonly used in before-after-designs. Comparing a mean of difference scores to a distribution of means of difference scores Population of measures at Time 1 and Time 2 Population of difference between measures at Time 1 and Time 2 Population of mean difference between measures at Time 1 and Time 2 (Whew!)

Paired Sample t Test Single-Sample Paired-Sample Single observation from each participant The observation is independent from that of the other participants Comparing a mean score to a distribution of mean scores . Two observations from each participant The second observation is dependent upon the first since they come from the same person. Comparing a mean of difference scores to a distribution of means of difference scores (I don’t make this stuff up)

Paired Sample t Test A distribution of scores. A distribution of differences between scores. Central Limit Theorem Revisited. If you plot the mean of randomly sampled observations, the plot will approach a normal distribution. This is true for scores and for differences between scores.

Difference Between Calculating Single-Sample t and Paired-Sample t Statistic Single Sample t Statistic Paired Sample t Statistic Standard Deviation of a Sample Standard Deviation of Sample Differences Standard Error of a Sample Standard Error of Sample Differences t Statistic for Single-Sample t Test T Statistic for Paired-Sample t Test

Paired Sample t Test Example We need to know if there is a difference in the salary for the same job in Boise, ID, and LA, CA. The salary of 6 employees in the 25th percentile in the two cities is given . Six Steps of Hypothesis testing for Paired Sample Test Profession Boise Los Angeles Executive Chef 53,047 62,490 Genetics Counselor 49,958 58,850 Grants Writer 41,974 49,445 Librarian 44,366 52,263 School teacher 40,470 47,674 Social Worker 36,963 43,542

Paired Sample t Test Example We need to know if there is a difference in the salary for the same job in Boise, ID, and LA, CA. Step 1: Define Pops. Distribution and Comparison Distribution and Assumptions Pop. 1. Jobs in Boise Pop. 2.. Jobs in LA Comparison distribution will be a distribution of mean differences, it will be a paired-samples test because every job sampled contributes two scores, one in each condition. Assumptions: the dependent variable is scale, we do not know if the distribution is normal, we must proceed with caution; the jobs are not randomly selected, so we must proceed with caution

Paired Sample t Test Example We need to know if there is a difference in the salary for the same job in Boise, ID, and LA, CA. Step 3: Determine the Characteristics of Comparison Distribution (mean, standard deviation, standard error) M = 7914.333 Sum of Squares (SS) = 5,777,187.333 Profession Boise Los Angeles X-Y D (X-Y)-M M = 7914.33 D^2 Executive Chef 53,047 62,490 -9,443 -1,528.67 2,336,821.78 Genetic Counselor 49,958 58,850 -8,892 -977.67 955,832.11 Grants Writer 41,974 49,445 -7,471 443.33 196,544.44 Librarian 44,366 52,263 -7,897 17.33 300.44 School teacher 40,470 47,674 -7,204 710.33 504,573.44 Social Worker 36,963 43,542 -6,579 1,335.33 1,783,115.11

Paired Sample t Test Example We need to know if there is a difference in the salary for the same job in Boise, ID, and LA, CA. Step 4: Determine Critical Cutoff df = N-1 = 6-1= 5 t statistic for 5 df , p < .05, two-tailed, are -2.571 and 2.571 Step 5: Calculate t Statistic Step 6 Decide

Independent t Test Compares the difference between two means of two independent groups. The comparison distribution is a difference between means to a distribution of differences between means. Population of measures for Group 1 and Group 2 Sample means from Group 1 and Group 2 Population of differences between sample means of Group 1 and Group 2

Independent t Test Paired-Sample Independent t Test Two observations from each participant The second observation is dependent upon the first since they come from the same person. Comparing a mean difference to a distribution of mean difference scores Single observation from each participant from two independent groups The observation from the second group is independent from the first since they come from different subjects. Comparing a the difference between two means to a distribution of differences between mean scores .

Independent t Test: Steps

Step 6 t Statistic for an independent-Samples t Test Step 7

Independent t Test Similar to previous steps except it takes more time to calculate the estimate of the standard error, called the pooled estimate of the standard error. Must calculate Pooled Variance, a weighted average of the estimates of the variance from both samples.

Standard Error of a Sample: estimates the Sample Error of the Population t Statistic for Single-Sample t Test Standard Deviation of a Sample: Estimates the Population Standard Deviation t Statistic for Independent t Test T Statistic for Paired-Sample t Test Degrees of Freedom for Single Sample t Test, and Paired Sample Variance for a sample

Degrees of Freedom for Independent Samples t Test Pooled Variance. Like adding together the weighted average of the variance from Variable X and Variable Y. Variance for a Distribution of means for Indep.-Samples t Test Variance for a Distribution of Differences between Means SD of the distribution of Differences Between Means

X1 X2 2 3 4 5 6 7