Standard Error of the Sample Means
 AKA Standard Error
 

•The mean, the variance, and the standard  deviation help estimate characteristics of the population from a single sample

•So if many samples were taken then the means of the samples would also form a normal distribution curve that would be close to the whole population.

•The larger the samples the closer the means would be to the actual value

•But that would most likely be impossible to obtain so use a simple method to compute the means of all the samples

 Probability Tests

  •What to do when you are comparing two samples to each other and you want to know if there is a significant difference between both sample populations 

•(example the control and the experimental setup) 

•How do you know there is a difference 

•How large is a “difference”? 

•How do you know the “difference” was caused by a treatment and not due to “normal” sampling variation or sampling bias?

Laws of Probability 

•The results of one trial of a chance event do not affect the results of later trials of the same event. p = 0.5 ( a coin always has a 50:50 chance of coming up heads) 

•The chance that two or more independent events will occur together is the product of their changes of occurring separately. (one outcome has nothing to do with the other) 

•Example: What’s the likelihood of a 3 coming up on a dice: six sides to a dice: p = 1/6 

•Roll two dice with 3’s p = 1/6 *1/6= 1/36 which means there’s a 35/36 chance of rolling something else… 

•Note probabilities must equal 1.0

•The probability that either of two or more mutually exclusive events will occur is the sum of their probabilities (only one can happen at a time). 

•Example: What is the probability of rolling a total of either 2 or 12? 

•Probability of rolling a 2 means a 1 on each of the dice; therefore p = 1/6*1/6 = 1/36 

• Probability of rolling a 12 means a 6 and a 6 on each of the dice; therefore p = 1/36 

•So the likelihood of rolling either is 1/36+1/36 = 2/36 or 1/18  

  

The Use of the Null Hypothesis

•Is the difference in two sample populations due to chance or a real statistical difference? 

•The null hypothesis assumes that there will be no “difference” or no “change” or no “effect” of the experimental treatment. 

•If treatment A is no better than treatment B then the null hypothesis is supported. 

•If there is a significant difference between A and B then the null hypothesis is rejected...  

T-test or Chi Square? Testing the validity of the null hypothesis

•Use the T-test (also called Student’s T-test) if using continuous variables from a normally distributed sample populations (ex. Height) 

•Use the Chi Square (X2) if using discrete variables (if you are evaluating the differences between experimental data and expected or hypothetical data)… Example: genetics experiments, expected distribution of organisms. 

 T-test

  •T-test determines the probability that the null hypothesis concerning the means of two small samples is correct 

•The probability that two samples are representative of a single population (supporting null hypothesis) OR two different populations (rejecting null hypothesis)

 

STUDENT’S T TEST

•      The student’s t test is a statistical method that is used to see if to sets of data differ significantly.

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•      The method assumes that the results follow the normal distribution (also called student's t-distribution) if the null hypothesis is true. 

•      This null hypothesis will usually stipulate that there is no significant difference between the means of the two data sets.

•      It is best used to try and determine whether there is a difference between two independent sample groups. For the test to be applicable, the sample groups must be completely independent, and it is best used when the sample size is too small to use more advanced methods.

•      Before using this type of test it is essential to plot the sample data from he two samples and make sure that it has a reasonably normal distribution, or the student’s t test will not be suitable.

•       It is also desirable to randomly assign samples to the groups, wherever possible.

EXAMPLE

•      You might be trying to determine if there is a significant difference in test scores between two groups of 

children taught by different methods. 

•      The null hypothesis might state that there is no significant difference in the mean test scores of the two 

sample groups and that any difference down to chance. 

The student’s t test can then be used to try and disprove the null hypothesis. 

RESTRICTIONS

•      The two sample groups being tested must have a reasonably normal distribution. If the distribution is skewed, then the student’s t test is likely to throw up misleading results. 

•      The distribution should have only one mean peak (mode) near the center of the group.

•      If the data does not adhere to the above parameters, then either a large data sample is needed or, preferably, a more complex form of data analysis should be used.

RESULTS

•      The student’s t test can let you know if there is a significant difference in the means of the two sample groups and disprove the null hypothesis.

•       Like all statistical tests, it cannot prove anything, as there is always a chance of experimental error occurring.

•      But the test can support a hypothesis. However, it is still useful for measuring small sample populations and determining if there is a significant difference between the groups.