Statistics T-test Black: pp 362-364 and 402-416

Example problem Is it possible that this sample is coming from a high school educated population?

Example problem METHOD B – Confidence interval from sample to population Is it possible that this sample is coming from a high school educated population? It is 95% likely that this sample is coming from a population with a mean between 9.2 and 11.6. Therefore this sample is probably not coming from a high school educated population.

Problem METHOD B What kinds of populations can this sample be coming from? Compute the sample mean. Compute the estimated standard deviation for the population Compute the SEM Find out 95% confidence interval for the population that generated this sample Conclude if the given population could have generated THIS sample?

INTODUCING METHOD C– t-test method Is it likely that this sample came from a population with a mean ____ ? Is the mean of this sample close enough to the population mean of ____ ? Testing the Null Hypothesis: The sample mean and the given population mean are equal. Alternative hypothesis: The sample mean is different from the population mean The sample mean is greater than (smaller than) the population mean ANSWER: One sample t-test

More on the one sample t-test Population parameter: It is FIXED. It does not change from sample to sample. Sample statistic: It VARIES. It changes from sample to sample. Test statistic: It has a KNOWN distribution and its probability of occurrence can be looked up. Estimate of the Variability in Sample statistic: How much does sample statistic change from sample to sample.

METHOD C: t-test method Is it likely that this sample came from a population with a mean ____ ? Is the mean of this sample close enough to the population mean of ____ ? Write the null and alternative hypotheses Calculate the sample mean Calculate the difference of sample mean from population mean Take the absolute value of the difference Estimate of the POPULATION SD Calculate the SEM Divide the difference of means by SEMTHIS IS THE T-STATISTIC Look up the probability of the t statistic using the excel function: TDIST(t-value, df, 2)

Example problem – t-test method Is it possible that this sample is coming from a high school educated population? The probability that this sample is coming from a population with a mean of 12 is only 2.9%. Therefore it is HIGHLY UNLIKELY that this sample is coming from a high school educated population.

METHOD C– t-test method for testing SPECIFIC hypotheses “Is it likely that this sample came from a population with a mean ____ ?” OR “Is the mean of this sample close enough to the population mean of ____ ?” Null Hypothesis: The sample mean and the given population mean are equal. Alternative Hypothesis: Option 1: The sample mean is different from the population mean  TWO TAILED TEST Option 2: The sample mean is greater than the population mean  ONE TAILED TEST Option 3: The sample mean is smaller than the population mean  ONE TAILED TEST Apply one sample t-test 

Example Problem 1 vs. 2-tailed tests Depression scores from a sample of men who just lost their jobs. Is the depression level in this sample normal? (mean level of depression in a normal population is 10)? Null Hyp: The sample is coming from a population with mean depression = 10 Alt Hyp: The sample is coming from a population with mean depression GREATER THAN 10

Example problem – t-test method Is the depression level in this sample normal?

Practice for 1 and 2 tails Is the sample of children from Hakkari coming from a population with normal level of anxiety? Do American Indian children have a normal level of language development? Do the 2006 entries to KU have a similar average GPA as the 2005 entries? Do rural women have the same mean age at marriage as the urban women? Is the post-diet weight of women who went on Weight-Watchers diet similar to the group of women who did not diet?

Doing a t-test We must find the probability of this t value: Depends on sample size (df=n-1) Depends on the alternative hypothesis TDIST(t,df, tails)

Example Is it likely that this sample is coming from a population with normal reading ability (mean of 100)?

Example Is it likely that this sample is coming from a population with normal reading ability (mean of 100)? Alternatively: Is the mean of this sample close enough to 100?

Example Is it likely that this sample is coming from a population with normal reading ability (mean of 100)? Alternatively: Is the mean of this sample close enough to 100? Null Hyp: The mean of this sample is not different from 100. Alternative Hyp: The mean of this sample is lower than 100.

Example Calculate the mean: 70.6

Example Calculate the mean: 70.6 Calculate the SD: 10.2

Example Calculate the mean: 70.6 Calculate the SD: 10.2 Calculate the SEM: 2.05

Example Calculate the mean: 70.6 Calculate the SD: 10.2 Calculate the SEM: 2.05 Calculate the t-value: 14.4 Calculate the probability of that t: TDIST(14.4, 24,1) = 0.000000

Example Calculate the mean: 70.6 Calculate the SD: 10.2 Calculate the SEM: 2.05 Calculate the t-value: 14.4 Calculate the probability of that t: TDIST(14.4, 24,1) = 0.000000 CONCLUSION?

Example – t-test Mean weight of golden retriever dogs are 42kg. A sample of 20 golden retrievers were fed a new flavor of Purina dog food, and their average weight was found to be 46kg. The SD was estimated to be 5 kg. Can we say that the new Purina flavor is not healthy (i.e. they eat too much, thus they get too fat) for golden retrievers?

What if I have two samples? Children from divorced families have average aggression scores of 67. Children from intact families have average aggression scores of 52. DO CHILDREN FROM DIVORCED FAMILIES HAVE HIGHER AGGRESSION SCORES THAN CHILDREN IN INTACT FAMILIES?

PROBLEM TYPE 2: Comparing means Is the achievement of children from divorced families different from that of children from intact families? Do boys have a higher level of aggression than girls? Are working women happier than women who stay at home? Are students at Koc more satisfied than students at Sabanci? Do children who attended kindergarten perform better in first grade than children who did not attend kindergarten? Do religious women have less authority in the family than non-religious women? Do students from affluent families have less independent decision making skills than students from modest family backgrounds?

? ? TWO SAMPLES: Population Sample Mean=x Mean=x Population Sample Mean=y Sample Mean=y ?

TWO SAMPLES: Population Mean=z Sample Mean=x ? Sample Mean=y ?

TWO SAMPLE t-test Are the two means “close enough”? What is “close enough”? Are the two samples coming from the same population?

“Close enough” has to be related to the variability of each sample mean Difference of means variability of the difference in means

“Close enough” has to be related to the variability of each sample mean Difference of means variability of the difference in means Variance of the difference = variance of first mean + variance of second mean SQRT(Variance of the difference) = SQRT (variance of first mean + variance of second mean)

Application Mean of sample 1 = 150 Mean of sample 2 = 170 Est SD of pop 1 = 34 Est SD of pop 2 = 32 N of sample 1 = 50 N of sample 2 = 50 Are the sample means significantly different from each other? Step 1: Null and Alternative Hypotheses Null Hyp: The means of the two samples are equal. OR: The two samples are COMING FROM THE SAME POPULATION.

Application - continued Step 2: SEM of each mean Population 1: 34/sqrt(50) = 4.81 Population 2: 32/sqrt(50) = 4.53 Step 3: t-statistic Numerator = 170-150=20 Denominator = Sqrt(4.812 + 4.532) = sqrt(23.14 + 20.52) = 6.61 t-statistic = 20/6.61 = 3.03 df = 50+50-2 = 98 P-value = 0.003 Step 4: conclusion Conclusion: The two groups have significantly different means. OR The two groups are NOT coming from the same population.

Example Sample A: Intact families Sample B: Divorced families Mean 108 102 SD 16 13 N 112 84

QUIZ 3 on Wed Oct 24