DIRECTIONAL HYPOTHESIS

1
DIRECTIONAL HYPOTHESIS

• The 1-tailed test
• Instead of dividing alpha by 2, you are looking
  for unlikely outcomes on only 1 side of the
  distribution
• No critical area on 1 sidethe side depends upon
  the direction of the hypothesis
• In this case, anything greater than the critical
  region is considered non-significant

-1.96 -1.65
0
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Non-Directional Directional Hypotheses

• Nondirectional
• Ho there is no effect
• (X µ)
• H1 there IS an effect
• (X ? µ)
• APPLY 2-TAILED TEST
• 2.5 chance of error in each tail
• Directional
• H1 sample mean is larger than population mean
• (X gt µ)
• Ho x µ
• APPLY 1-TAILED TEST
• 5 chance of error in one tail

-1.96 1.96
1.65
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Why we typically use 2-tailed tests

• Often times, theory or logic does allow us to
  prediction direction why not use 1-tailed
  tests?
• Those with low self-control should be more likely
  to engage in crime.
• Rehabilitation programs should reduce likelihood
  of future arrest.
• What happens if we find the reverse?
• Theory is incorrect, or program has the
  unintended consequence of making matters worse.

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STUDENTS t DISTRIBUTION

• We cant use Z distribution with smaller samples
  (Nlt100) because of large standard errors
• Instead, we use the t distribution
• Approximately normal beginning when sample size gt
  30
• Probabilities under the t distribution are
  different than from the Z distribution for small
  samples
• They become more like Z as sample size (N)
  increases

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THE 1-SAMPLE CASE

• 2 Applications
• Single sample means (large Ns) (Z statistic)
• May substitute sample s for population standard
  deviation, but then subtract 1 from n
• s/vN-1 on bottom of z formula
• Smaller N distribution (t statistic), population
  SD unknown

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STUDENTS t DISTRIBUTION

• Find the t (critical) values in App. B of Healey
• degrees of freedom
• of values in a distribution that are free to
  vary
• Here, df N-1
• When finding t(critical) always use lower df
  associated with your N
• Practice
• ALPHA TEST N t(Critical)
• .05 2-tailed 57
• .0 1 1-tailed 25
• .10 2-tailed 32
• .05 1-tailed 15

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Example Single sample means, smaller N and/or
unknown pop. S.D.

• A random sample of 26 sociology grads scored an
  average of 458 on the GRE sociology test, with a
  standard deviation of 20. Is this significantly
  higher than the national average (µ 440)?
• The same students studied an average of 19 hours
  a week (s6.5). Is this significantly different
  from the overall average (µ 15.5)?
• USE ALPHA .05 for both

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1-Sample Hypothesis Testing (Review of what has
been covered so far)

• If the null hypothesis is correct, the estimated
  sample statistic (i.e., sample mean) is going to
  be close to the population mean
• 2. When we set the criteria for a decision, we
  are deciding how far the sample statistic has to
  fall from the population mean for us to decide to
  reject H0
• Deciding on probability of getting a given sample
  statistic if H0 is true
• 3 common probabilities (alpha levels) used are
  .10, .05 .01
• These correspond to Z score critical values of
  1.65, 1.96 258

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1-Sample Hypothesis Testing (Review of what has
been covered so far)

• 3. If test statistic we calculate is beyond the
  critical value (in the critical region) then we
  reject H0
• Probability of getting test stat (if null is
  true) is small enough for us to reject the null
• In other words There is a statistically
  significant difference between population
  sample means.
• 4. If test statistic we calculate does not fall
  in critical region, we fail to reject the H0
• There is NOT a statistically significant
  difference

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2-Sample Hypothesis Testing (intro)

• Apply when
• You have a hypothesis that the means (or
  proportions) of a variable differ between 2
  populations
• Components
• 2 representative samples Dont get confused
  here (usually both come from same sample)
• One interval/ratio dependent variable
• Examples
• Do male and female differ in their aggression (
  aggressive acts in past week)?
• Is there a difference between MN WI in the
  proportion who eat cheese every day?
• Null Hypothesis (Ho)
• The 2 pops. are not different in terms of the
  dependent variable

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2-SAMPLE HYPOTHESIS TESTING

• Assumptions
• Random (probability) sampling
• Groups are independent
• Homogeneity of variance
• the amount of variability in the D.V. is about
  equal in each of the 2 groups
• The sampling distribution of the difference
  between means is normal in shape

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2-SAMPLE HYPOTHESIS TESTING

• We rarely know population S.D.s
• Therefore, for 2-sample t-testing, we must use 2
  sample S.D.s, corrected for bias
• Pooled Estimate
• Focus on the t statistic
• t (obtained) (X X)
• s x-x
• were finding the
• difference between the two means
• and standardizing this difference with the
  pooled estimate

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2-SAMPLE HYPOTHESIS TESTING
2-Sample Sampling Distribution difference
between sample means (closer sample means will
have differences closer to 0)

• t-test for the difference between 2 sample means
• Addresses the question of whether the observed
  difference between the sample means reflects a
  real difference in the population means or is due
  to sampling error

-2.042 0 2.042
ASSUMING THE NULL!
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Applying the 2-Sample t Formula

• Example
• Research Hypothesis (H1)
• Soc. majors at UMD drink more beers per month
  than non-soc. majors
• Random sample of 205 students
• Soc majors N 100, mean16, s1.0
• Non soc. majors N 105, mean15, s0.9
• Alpha .01
• FORMULA
• t(obtained) X1 X2
• pooled estimate

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Answers

• Null hypothesis
• There is no difference in mean number of fights
  between inmates with tattoos and inmates without
  tattoos.
• Use a 1 or 2-tailed test?
• One-tailed test because the theory predicts that
  inmates with tattoos will get into MORE fights.

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Answers

• Calculations
• Obtained value
• Reject the null?
• Yes because the t(obtained) (19.09) is greater
  than the t(critical, one-tail, df398) (1.658)
• This t value indicates there are 19.09 standard
  error units that separate the two mean values
• VERY unlikely we got this big a difference due to
  sampling error
• Research hypothesis restated as non-directional
• There is a difference in the mean number of
  fights reported by inmates with tattoos and
  inmates without tattoos.
• Would you come to a different conclusion if you
  used a 2-tailed test?
• No, because 19.09 is still well beyond the
  2-tailed critical value (1.980).

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2-Sample Hypothesis Testing in SPSS

• Independent Samples t Test Output
• Testing the Ho that there is no difference in
  number of adult arrests between a sample of
  individuals who were abused/neglected as children
  and a matched control group.

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Interpreting SPSS Output

• Difference in mean of adult arrests between
  those who were abused as children control group

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Interpreting SPSS Output

• t statistic, with degrees of freedom

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Interpreting SPSS Output

• Sig. (2 tailed)
• gives the actual probability of making a Type I
  (alpha) error
• a.k.a. the p value p probability

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Sig. Probability

• Number under Sig. column is the exact
  probability of obtaining that t-value (finding
  that mean difference) if the null is true
• When probability gt alpha, we do NOT reject H0
• When probability lt alpha, we DO reject H0
• As the test statistics (here, t) increase, they
  indicate larger differences between our obtained
  finding and what is expected under null
• Therefore, as the test statistic increases, the
  probability associated with it decreases

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Example 2 Education Ageat which First Child
is Born
H0 There is no relationship between whether an
individual has a college degree and his or her
age when their first child is born.
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Education Age at which First Child is Born

1. What is the mean difference in age?
2. What is the probability that this t statistic is
   due to sampling error?
3. Do we reject H0 at the alpha .05 level?
4. Do we reject H0 at the alpha .01 level?