Assignment

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Assignment

• After this lecture, start Assignment 4.
• Its in the course binder.

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Compared to what?

• Lecture 10
• Hypothesis test for a mean

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Overview

• What is a hypothesis?
• How do you compare a hypothesis to your data
• using confidence intervals
• using sampling distributions

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1.

• What is a hypothesis?

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What is a hypothesis?

• Hypothesis Claim about a parameter
• Today claim about the population mean mY.
• E.g., the average starting salary of all
  sociology BAs
• Two alternatives
• null hypothesis (H0)
• a neutral claim
• research hypothesis (H1)
• one-tailed or two-tailed

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Two-tailed hypotheses

• Does journaling affect depression
• Effect has two possible directions
• Journaling works through issues?depression lifts
• Journaling dwells on problems?depression deepens
• Let mY be average change in depression after
  journaling.
• Null hypothesis journaling does not affect
  depression
• H0 mY 0
• Research hypothesis journaling affects
  depression

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One-tailed hypotheses

• Do smokers have lighter babies?
• Expected effect has one direction
• In general population,
• average birthweight is say 7 pounds
• Let mY be average weight of smokers children.
• Research hypothesis smokers have lighter babies
• H1 mY lt 7
• Null hypothesis smokers have average babies (or
  even heavy babies)
• H0 mY 7 (or H0 mY gt 7)
• both forms are useful

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One tail or two?

• Key Does the research hypothesis specify
  direction?
• One tail
• H1 mY lt 7
• H0 mY 7 (or H0 mY gt 7)
• Two tails

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2.

• Hypothesis tests

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Hypothesis test Definition

• Given evidence from a sample,
• we test (decide whether to reject)
• the null hypothesis
• about the population

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Research question

• Should you finish college?
• Should you delay graduation for part-time work?
• UPS offers 9 an hour plus 3000 a year for books
  and tuition
• 9/hour X 2000 hours/year 3000 21,000 per
  year
• Will you do better than that when you graduate?

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Null vs. research hypothesisSymbols

• Let Y be the starting salary for a sociology BA
• mY is the average for the population of sociology
  BAs
• H1 mY gt 21K
• The average is more than what you could make at
  UPS.
• one tail or two?
• H0 mY lt 21K
• The average is no more than what you could make
  at UPS.
• (Note Your book would say H0 mY 21K instead
  of mY lt 21K)

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Null vs. research hypothesisPicture

• H0 mY lt 21K
• H1 mY gt 21K

H1
H0
mY
21K
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Sample pertinent data

• National Association of Colleges and Employers
• Sample of N92 Sociology BAs, graduating 2000-01
• Variable starting salary in thousands
• Cases 38.0, 28.0, 28.0, 24.6,
• The sample mean is greater than 21K
• But the hypothesis asks about the population mean
• which is probably different (sampling error)

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2a.

• Hypothesis testConfidence interval method

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Confidence interval for the meanReview

• Lecture 8, Example 2
• t confidence interval
• We are 95 sure
• that mY is between 27,369 and 30,299
• Confidence intervals like this
• fail to contain mY
• only 5 of the time.

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Confidence interval and hypotheses

• H0 mY lt 21K
• H1 mY gt 21K

H1
H0
mY
CI
21K
30K
27K
CI mY is in 27,369-30,299
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Rejecting H0

• The confidence interval (CI)
• contains plausible values of mY
• One of these plausible values is right
• in 95 of all samples
• None of the values in H0
• are in the CI
• So we reject H0
• as implausible

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Hypothesis testsConfidence interval method

• Draw, on a number line,
• the hypotheses
• the confidence interval
• If the confidence interval overlaps H0
• then H0 is plausible
• you dont reject H0
• If the confidence interval doesnt overlap H0
• then H0 is implausible
• you reject H0

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Interpretation

• We rejected the idea (H0)
• that new sociology BAs
• make no more, on average, than students at UPS
• We accepted the idea (H1)
• that new sociology BAs
• make more, on average, than students at UPS

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2b.

• Hypothesis testSampling distribution method

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Hypothesis testSampling distribution method

• Assuming, provisionally, that H0 is true
• Draw what the sampling distribution would look
  like
• Where is your sample in this distribution?
• If your sample looks extreme (unusual)
• then the sampling distribution is implausible
• so you reject H0

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1. Assume, provisionally, that H0 is true

• Suppose H0 mY 21K is true
• Note in this method, we dont say H0 mY lt 21K

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2. Draw the sampling distribution

• If H0 were true, then
• across all possible samples
• of size N92
• would have
• mean
• and standard deviation
• a.k.a. standard error

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2. Draw the null sampling distribution

• If H0 were true, then this would be the sampling
  distribution
• called the null sampling distribution

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3. Where would our sample fall in the null
distribution?

• Our sample would look
• extreme
• improbable

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4. Conclusion

• When we assumed H0 was true
• our sample looked extreme and improbable
• in the null sampling distribution
• Maybe our sample really is extreme and improbable
• But more likely H0 is false
• We reject H0

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Interpretation (again)

• We rejected the idea (H0)
• that new sociology BAs
• earn, on average,
• no more than students at UPS
• We accepted the idea (H1)
• that new sociology BAs
• earn, on average, more than students at UPS

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The t statistic

• We dont really look at the sampling distribution
  of
• We look at the sampling distribution of t
• t is the standardized sample mean
• The number of standard errors that separate our
  sample mean from the population mean under H0

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The t statistic

• In our sample, N92, Ybar28.834, SY7.095
• If H0 mY 21 were true,
• then Ybar would be
• t estimated standard errors from mY

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3. Draw the null sampling distribution (again)

• If H0 is true, then
• across all possible samples
• of size N92
• t follows
• a t distribution
• with dfN-191 degrees of freedom
• This is much like the normal distribution
• So any t score greater than 2 will look extreme

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3. Where is our sample in the null distribution?

• Our sample looks
• extreme
• improbable

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4-5. Conclusion and interpretation(again)

• When we assumed H0 was true
• our sample looked extreme and improbable
• So we reject H0
• We reject the idea (H0)
• that new sociology BAs earn, on average,
• no more, on average, than students at UPS
• We accept the idea (H1)
• that new sociology BAs
• earn more, on average, than students at UPS

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Extreme? Improbable?

• Under H0, its obvious our sample is
• extreme
• improbable
• But how extreme is extreme?
• How improbable is improbable?

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How improbable?The p value

• If H0 were true
• only one out of 10 quintillion samples
• would have a t statistic of at least 10.59
• which we got
• This is the p value p10-16 (1 in 10
  quintillion)
• the probability of a sample as extreme as ours
• if H0 is true
• Typically we reject H0
• if plt.05 (or plt.01)
• .05 (or .01) is a conventional significance level
  (a)

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Estimating a p value

• How do you know if your p value is lt .05 (or
  .01)?
• SPSS tells you p (sig.)
• Excel can take t and give you p
• TDIST(t,df,tails)
• e.g., TDIST(10.59,91,1) should give you 10-16
• It doesnt, quite. But its close.
• What if youre not using a computer?
• You can estimate p from the zand t table

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Reading a t table

• Remember!
• We have a one-tailed research hypothesis
• So we read the table for a one-tailed test
• We dont see dfN-191, so we use the closest
  value
• df100

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Reading a t table

• If t2.36 then p.010
• If t2.63 then p.005
• If t3.39 then p.0005
• Our t10.59
• Since larger ts have smaller ps
• we know plt.0005
• We typically reject H0 when p lt .05 (or .01)
• So we reject H0 here

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4-5. Conclusion and interpretation(using p value)

• If H0 were true
• if new sociology BAs had an average salary of
  21K
• then wed see a sample as rich as this one
• in less than 0.05 of all samples
• I.e., plt.0005
• So we reject H0 in favor of H1
• We think new sociology BAs average higher than
  21K

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Summary Hypotheses and hypothesis tests

• Hypothesis claim about a parameter
• research hypothesis vs. null hypothesis(H1 vs.
  H0)
• research hypothesis can be
• one-tailed
• or two-tailed
• Hypothesis test
• using a sample to evaluate H1 and H0

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Summary Confidence interval method

• On a line representing parameter values (mY)
• Draw the hypotheses (H1 vs. H0)
• Draw the confidence interval (CI)

• If CI overlaps H0
• Dont reject H0 (its plausible)
• Otherwise
• Reject H0 (its implausible)

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Summary Sampling distribution method

• Assume H0 is true
• draw the sampling distribution of t
• Calculate t for the sample
• and draw it

• If t is extreme/improbablei.e., if p is small
• Reject H0 (its implausible)
• Otherwise
• Dont reject H0 (its plausible)

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Next topic

• Hypothesis test for a proportion