Differences Between Means t test

1
Differences Between Meanst test

• Much of the activity of the researcher is
  directed toward comparing the performance of two
  groups.
• A t test is a procedure that compare the scores
  of some dependent variable from two different
  groups.
• For example, a teacher wishes to determine
  whether there are differences between girls and
  boys achievement in reading. To answer this
  question the teacher will

2

• Draw a random sample of first-grade girls and a
  random sample of first-grade boys from a large a
  school district.
• The average reading achievement of both groups on
  a standardized test is
• Girls m 50 and boys m 46
• The results suggest that girl on the average,
  have higher achievement in reading.
• It is possible that the difference is due only to
  errors, sampling errors. The population means
  for girls and boys are identical
• This possibility is known as null hypothesis

3
Null Hypothesis

• The Null-Hypothesis simply states that there is
  no difference between the means of the two
  populations from which we drew our two samples.
  The true difference between the means (in the
  population) is zero.
• H0 ??????2 0
• Where
• H0 is the symbol for the null hypothesis
• ???is the symbol for the population mean for one
  group
• H1 ??????2 (in this case is a directional
  hypothesis)
• Where
• H1 is the symbol for an alternative hypothesis
• ???is the symbol for the population mean for one
  group

4
Example 2

• An investigator wished to determine whether there
  are differences between men and women voters in
  their attitudes toward welfare.
• Samples of men and women were drawn at random and
  administered an attitude scale to obtain a score
  for each subject.
• Women had a mean of 40 (on a scale 0 to 50, where
  50 is the most favorable). Men had a mean of 35.
  
• The researcher wishes to determine whether there
  is a significant difference between mean and
  women.
• What accounts for the 5 point difference?
• One possible explanation is the null hypothesis,
  which states that there is no true difference
  between men and women - that the observed
  difference is due to sampling error created by
  random sampling.

5

• Example 3
• A random sample of kittens is fed a vitamin
  supplement from birth to see if the supplement
  increases their visual acuity.
• Another random sample is fed a placebo that looks
  like the supplement but contains no vitamins.
• At the end of the study, both samples are tested
  for visual acuity an average acuity score is
  calculated for each sample.
• Those that took the supplement scored 4 points
  higher than the control group.
• What accounts for the 4 point difference?
• One possible explanation is the null hypothesis,
  which states that there is no true difference
  between the two samples of kittens - that the
  observed difference is due to sampling errors
  created by random sampling.

6
What leads the t-test to give us a low
probability that the null hypothesis is correct?

• Three basic factors
• The larger the samples, the less likely the
  difference between two means was created by
  sampling error.
• The larger the observed difference between the
  two means, the less likely that the difference
  was created by sampling errors.
• The smaller the variance among the subjects, the
  less likely that the difference between two means
  was created by sampling errors and the most
  likely the null hypothesis was rejected.

7
Assumptions underlying the t-test

• The scores must be interval or ratio in nature
• The scores must be measures on random samples
  from the respective populations
• The populations from which the samples were drawn
  must be normally distributed
• The populations from which the samples were drawn
  must have approximately the same variability
  (homogeneity of variance)

8
Types of t-tests

• Independent data (sometimes called uncorrelated
  data)
• Dependent data (sometimes called correlated data)
  
• Means that result from this type of t test are
  subject to less error than for independent data
• the matching subjects assures us that the two
  groups are more similar than if just any two
  independent samples were used.

9

• Example of dependent t-tests
• In a study of visual acuity, same-sex siblings
  (two brothers or two sisters) were identified for
  a study.
• For each pair of siblings, a coin was tossed to
  determine which one received a vitamin supplement
  and which received a placebo.
• For each subject in the experimental group, there
  is a same sex-sibling in the control group.

10
How to calculate t- test for independent samples

• Formula when the population standard error is
  known
• Formula for t-test (standard error of sample)

11
Calculating Standard Error of the Difference
Between Means

• Population standard deviation is known
• Sample standard deviation

12
Reporting the results of t Tests

• The difference between the means is statistically
  significant (t 3.22, df 10, p
• The difference between the means is significant
  at the .01 level (t 3.22, df 10)
• The symbol for probability is a lower-case p.
  Thus, if we find that the probability that the
  null hypothesis is true in a given study is less
  that 5 in 100, this result would be expressed as
  p

13
Degree of Freedom

• The number of observations that are free to vary
• In calculation of t test df is N - 1 for both
  samples
• (N1 - 1) (N2 - 1)
• If you have 20 students in one group and 10 in
  another group the df 28

14
Reporting the results of t-tests
15
Errors in making decisions

• Type I error is the error of rejecting the null
  hypothesis when it is true.
• Type II error is the error of failing to reject
  the null hypothesis when it is false.

16
One-tailed and two-tailed t tests

• Two-tailed tests means that the investigator
  proposes a null hypothesis that ??????2, so, if
  he or she decides to reject the hypothesis, it
  may be either ??????2 or ??????1. The null
  hypothesis will be rejected if the t value (or z,
  or other statistics) is either to the extreme
  left of the sampling distribution of to the
  extreme right.
• One-tailed test means that the investigator
  predicts before collecting the data that ???is
  greater than ?2, the alternate hypothesis is not
  ??????2, but ??????2. They are stating a
  directional hypothesis.