Hypothesis testing

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Hypothesis testing Inferential Statistics
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Hypothesis Testing

• The process of determining whether a hypothesis
  is supported by the results of the study.
• Use inferential statistics to draw conclusions
  (make inferences) about the population based on
  data collected from a sample.
• Ex cholesterol levels while on an all fruit
  diet vs. no diet at all.
• Hypothesis individuals in the all fruit diet
  will have lower levels of cholesterol than
  general population.

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Null and Alternate Hypothesis

• Goal of science is to reject untrue information,
  thereby the information that is left over is
  assumed to be true.
• Statistically impossible to show that something
  is true.
• Statistically possible to show that something is
  false.
• Counter intuitive rationale
• we must reject a hypothesis as false in order to
  find support for the hypothesis we are seeking.

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Null Hypothesis

• Ex we want to show that an all fruit diet lowers
  cholesterol compared to no diet at all.
• Null hypothesis no difference between the groups
  being compared.
• H0 µ 0 µ 1
• H0 µ of all fruit µ of general population
• Goal of research study is to determine whether a
  null hypothesis is true or false.
• If the null hypothesis is rejected, then a
  possible true result remains.

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Alternate Hypothesis

• Ex we want to show that an all fruit diet lowers
  cholesterol compared to no diet at all.
• Alternate (or research) hypothesis there is a
  significant difference between the groups being
  tested.
• Ha µ 0 lt µ 1
• Ha µ of all fruit lt µ of general population
• Goal of research study is to support the
  alternate hypothesis by rejecting the null
  hypothesis.

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Directional hypothesis or One-tailed

• Directional hypothesis (one-tailed hypothesis)
• The experimenter predicts the direction of the
  expected results.
• Alternate directional hypothesis
• Ha µ 0 lt µ 1
• Ha µ of all fruit lt µ of general population
• Null directional hypothesis H0 µ 0 µ 1
• H0 µ of all fruit µ of general population

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Nondirectional hypothesis or Two-tailed

• Nondirectional hypothesis (two-tailed hypothesis)
• The experimenter predicts differences between the
  groups, but is unsure what the differences will
  be.
• Alternate directional hypothesis Ha µ 0 ? µ 1
• Ha µ of all fruit ? µ of general population
• Null directional hypothesis H0 µ 0 µ 1
• H0 µ of all fruit µ of general population

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Type I Error

• Sometimes we can make mistakes in our research.
• At times, we reject the null hypothesis when it
  should NOT have been rejected.
• Ex we say the all fruit diet lowers cholesterol
  when it actually does not.
• Also known as a false positive we say there was
  a difference between groups, when in reality
  there was no difference.
• We found a difference, but it was due to chance
  and the results are a fluke.

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Type II Error

• In some cases, the null hypothesis SHOULD be
  rejected, but we fail to find a difference
  between the groups.
• The null hypothesis is false, but accept it
  anyway.
• Ex we say the 2 groups have equal levels of
  cholesterol when in fact the all-fruit group has
  lower levels.
• Somehow, we failed to find a difference between
  the 2 groups.

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Type I vs. Type II errors

• Type I error
• Null hypothesis not pregnant
• Alternative hypothesis pregnant
• If we reject the null (not pregnant) but we are
  incorrect, she is going to think she is pregnant
  when she is really not.
• Leads to actions
• Freaking out
• Happiness tell friends
• Buy baby clothes

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Type I vs. Type II errors

• Type II error
• Null hypothesis not pregnant
• Alternative hypothesis pregnant
• if we fail to reject the null, we keep the null
  and she will think she is not pregnant when she
  really is,
• Leads to no actions
• no prenatal care
• May go drinking and hurt
• the fetus.

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Type I vs. Type II

• Which is worse for research?
• Type I saying a result is true when it is not
  true is more detrimental to research.
• But in other cases, maybe Type II can be worse.
• In order to avoid both errors, researchers try to
  replicate the results.

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Statistical Significance

• Ex we find that the all-fruit diet group has
  lower levels of cholesterol than the rest of the
  population.
• Significantly lower!
• Statistical significance at .05 level (p .05)
• We will get these results by chance only 5 times
  or less out of 100.
• 95 of the time, these results are due to our
  manipulation
• We can reject the null hypothesis because the
  pattern of the data are unlikely to have occurred
  by chance.
• probability of making a Type I error 5 times or
  less out of 100
• The field has established the alpha level at .05

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Single-group design

• Research involving only one group and no control
  group.
• Simplest kind of hypothesis testing
• We compare the results of the group (sample) with
  the performance of the general population.
• Use parametric tests, a type of inferential
  statistics, that requires certain parameters
  about the population (i.e., mean, standard
  deviation)
• t test

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Single sample t-test

• Parametric inferential statistical test of the
  null hypothesis.
• Used for a single sample when we know the
  population mean, but not the population standard
  deviation.
• T-tests compare differences between the mean of a
  sample and the mean of a population.
• However, we need to compare the sample mean with
  a distribution of sample means

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Sampling Distribution

• Distribution of sample means based on random
  samples, of fixed sizes, from a population.
• Ex IQ scores of 1000 people (i.e., population)
• Take 100 samples of 10 people, plot the mean IQ
  of each sample.
• The mean of the population IQ will equal the mean
  of the distribution of means.

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Sampling Distribution

• Distribution of scores (pop.) and distribution of
  means have the same mean.
• The standard deviation (SD) of distribution of
  scores is bigger than the SD distribution of
  means.

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Sampling Distribution

• The standard deviation of a distribution of
  sample means has a new name.
• Standard error of the mean the standard
  deviation of a distribution of means.
• sM s
• vN
• Where

• s ?( X - X ) ²
• _________
• N - 1

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T-test

• To calculate a t-test, we need to know the sample
  mean, the population mean, and the standard error
  of the mean.
• t (X - µM)
• sM
• s 2.97
• Pop. mean 11
• t 1.05
• Our sample mean falls 1.05 standard deviations
  above the pop. mean.
• Is this difference large enough to be
  statistically significant?

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T-distributions

• We compare the t-value to a standardized
  distribution of t-values (i.e., t distributions).
• As the sample size increases, the t-distributions
  approaches a normal distribution.
• In t-distributions, we base sample size in terms
  of degrees of freedom or df

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Degrees of freedom

• Degrees of freedom
• Number of scores in a sample that are free to
  vary.
• Ex 2, 5, 6, 9, 11, 15 mean 8
• In order to maintain the mean of 8, five scores
  are free to vary, except for the last one.
• df N - 1

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T-distributions
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One-tailed t-Test

• We compare our t-value to a distribution of
  t-values.
• If our t-value is larger than the cut-off, we
  reject the null hypothesis.
• Directional hypothesis
• Ha µ of all fruit lt µ of general population
• H0 µ of all fruit µ of general population

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Two-tailed t-test

• If t-value falls in the region of rejection, we
  reject the null hypothesis.
• Nondirectional hypothesis
• Ha µ of all fruit ? µ of general population
• H0 µ of all fruit µ of general population

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T-table (appendix A.3)

• Which test is more conservative, one-tailed or
  two-tailed?
• Two-tailed test because it is more difficult to
  beat the critical value to reject the null
  hypothesis.

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Statistical Power

• Probability that we can reject the null
  hypothesis and find significant differences when
  differences truly exist.
• To increase power
• Use one-tailed tests
• Increase the sample size as sample size
  increases, the critical value decreases (i.e.,
  easier to reject the null hypothesis when the
  null is false)

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Steps to hypothesis testing

• The t test
• The six steps of hypothesis testing
• 1. Identify mean of sample and mean of population
  of sample means.
• 2. State the hypotheses (null and alternate)
• 3. Characteristics of the comparison distribution
• Find standard error of the mean
• 4. Critical values
• 5. Calculate t-value
• 6. Decide to reject or sustain the null
  hypothesis.