Testing statistical hypotheses about when is not known: the one sample ttest

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Testing statistical hypotheses about ?? when ??
is not known the one sample t-test

• Minium, Clarke Coladarci, Chapter 13

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Testing statistical hypotheses about ?? when ??
is not known

• We have made use of the z distribution (the
  standard normal distribution) to do hypothesis
  testing and to compute confidence intervals for
  example
• For hypothesis testing we can calculate
• and determine if Z exceeds our z-score cutoff
  (e.g., 1.64)
• For confidence intervals we can calculate
• Confidence interval Mean 1.96( )
• These two calculations require knowing ? so that
  we can calculate
• What if we didnt know ? and had to estimate it
  from our sample?
• What should we use to estimate ???
• Would we be able to substitute this estimate
  (estimated ) into our formulas and have
  everything work out the same way?

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Sampling Distributions -- biased and unbiased
estimates

• We might think that ?? can be estimated by
• but it turns out that this is not quite the
  case.
• S is said to be a biased estimator of ?
• We know that if you repeatedly choose samples of
  size n and compute the mean ( ) for each
  sample, we will create the sampling distribution
  of means.
• The mean of the sampling distribution of ( )
  will equal that of the original distribution (?).
• In this sense the sample mean ( ) is an
  unbiased estimate of ? because on average
  ??

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Sampling Distributions -- biased and unbiased
estimates

• If we repeatedly choose samples of size n and
  compute the variance (S2) for each sample,
• we will create a distribution of variances.
• The mean of the distribution of variances will
  not equal the variances of original distribution
  (?2).
• In this sense the sample variance (S2) is a
  biased estimate of ?2 because on average S2 ? ?2 .

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Sampling Distributions -- biased and unbiased
estimates

• But, if we repeatedly choose samples of size n
  and compute the the variance as
• we will create a distribution of variances and
  the mean of the distribution of s2 (computed with
  n-1 in the denominator) will equal of the
  variance of original distribution (?2).
• We refer to n-1 as the number of degrees of
  freedom (df) i.e., df n-1
• In this sense the sample variance computed with
  n-1 in the denominator (s2) is an unbiased
  estimate of ?2 because on average s2 ?2.

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Sampling Distributions -- biased and unbiased
estimates

• Therefore, going back to our first question,
• What should we use to estimate ?
• The answer is we should first compute an unbiased
  estimate of ?2 as
• And then we should compute an estimate of as
• where
• NOTE Be able to
• explain the difference between s2 and S2
• explain the difference between s and
• explain the difference between and

Web Demo
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The t-distribution

• What we know already
• The distribution of sample means has the
  following parameters
• Thus any particular mean can can be converted to
  a z score in the following way
• producing the standard normal distribution or,
  in other words, a sampling distribution of
  z-scores.

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The t-distribution

• Gosset raised the following question
• How would things change if we divided by
  rather than itself?
• Would the resulting sampling distribution be
  normal?
• The answer is no in general t-scores are not
  normally distributed.
• But the distribution of t-scores is symmetric
• and has a mean of zero
• The exact form of the t-distribution depends on n
• For large n the t-distribution approaches the
  standard normal distribution (the z-distribution)
  but for small n the t-distribution becomes
  leptokurtic.

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The t-distribution
The good news is that the t-distribution has a
definite form we know this because of smart
people like Gossett
(You dont have to remember this)
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The t-distribution
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The t-distribution
We use the t-distribution in same way we use the
z-distribution we find t? in the same way we
find z?
t?
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The one sample t-test

• Consider our baby/supercharged vitamin example
  again
• Lets say you know that on average the first
  steps are taken at 14 months
• but you dont know ?

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The one sample t-test

• Assume youve chosen a sample of 16 babies and
  given them vitamins that you expect will speed
  their development and hence lead them to walk at
  an earlier age.
• H0 ?V ?NV
• H1 ?V lt ?NV
• Set ? .05 and perform a one tailed t-test.
• The comparison distribution is the t-distribution
  
• with df (n - 1) 15, and t? -1.753
• Find the mean ( ) and standard deviation (s)
  of the sample

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The one sample t-test

• The sample mean is 12 and and standard
  deviation s 3
• Calculate s/sqrt(16) 3/4 .75
• Calculate (12-14)/.75 -2.67
• t -267, which is less than t? -1.753
• Therefore, reject H0.
• Conclude that the vitamins had a statistically
  significant effect.

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The one sample t-test

• Another Example?

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Confidence intervals when ? is estimated by s

• Confidence intervals about a mean
• If sample mean is 12 and, the standard
  deviation s 3 and n 16, then
• Calculate s/sqrt(16) 3/4 .75
• There are 16-1 15 dfs
• t? 2.132 (for a 95 CI)
• CI 12 2.132(.75) 10.401 to 13.599

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Additional Points

• Assumption of Population Normality
• Sample t-ratios follow the t-distribution exactly
  only if the samples have been randomly selected
  from a population of observations that itself has
  the normal shape
• Levels of significance versus p values
• We can use the t-table to set our cutoff values
  but it does not provide the exact p-value for all
  possible t-values for all dfs.
• However, computer programmes can do this.
• e.g., the tdist function which is analogous to
  the normdist function
• TDIST(x, degrees_freedom, tails)
• X is the numeric value at which to evaluate the
  distribution.
• Degrees_freedom is an integer indicating the
  number of degrees of freedom.
• Tails specifies the number of distribution
  tails to return. If tails 1, TDIST returns the
  one-tailed distribution. If tails 2, TDIST
  returns the two-tailed distribution.