The one sample ttest

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The one sample t-test

• November 14, 2006

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From Z to t

• In a Z test, you compare your sample to a known
  population, with a known mean and standard
  deviation.
• In real research practice, you often compare two
  or more groups of scores to each other, without
  any direct information about populations.
• Nothing is known about the populations that the
  samples are supposed to come from.

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The t Test for a Single Sample

• The single sample t test is used to compare a
  single sample to a population with a known mean
  but an unknown variance.
• The formula for the t statistic is similar in
  structure to the Z, except that the t statistic
  uses estimated standard error.

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From Z to t
Note lowercase s.
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Degrees of Freedom

• The number you divide by (the number of scores
  minus 1) to get the estimated population variance
  is called the degrees of freedom.
• The degrees of freedom is the number of scores in
  a sample that are free to vary.

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Degrees of Freedom

• Imagine a very simple situation in which the
  individual scores that make up a distribution are
  3, 4, 5, 6, and 7.
• If you are asked to tell what the first score is
  without having seen it, the best you could do is
  a wild guess, because the first score could be
  any number.
• If you are told the first score (3) and then
  asked to give the second, it too could be any
  number.

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Degrees of Freedom

• The same is true of the third and fourth scores
  each of them has complete freedom to vary.
• But if you know those first four scores (3, 4, 5,
  and 6) and you know the mean of the distribution
  (5), then the last score can only be 7.
• If, instead of the mean and 3, 4, 5, and 6, you
  were given the mean and 3, 5, 6, and 7, the
  missing score could only be 4.

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The t Distribution

• In the Z test, you learned that when the
  population distribution follows a normal curve,
  the shape of the distribution of means will also
  be a normal curve.
• However, this changes when you do hypothesis
  testing with an estimated population variance.
• Since our estimate of ? is based on our sample
• And from sample to sample, our estimate of ? will
  change, or vary
• There is variation in our estimate of ?, and more
  variation in the t distribution.

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The t Distribution

• Just how much the t distribution differs from the
  normal curve depends on the degrees of freedom.
• The t distribution differs most from the normal
  curve when the degrees of freedom are low
  (because the estimate of the population variance
  is based on a very small sample).
• Most notably, when degrees of freedom is small,
  extremely large t ratios (either positive or
  negative) make up a larger-than-normal part of
  the distribution of samples.

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The t Distribution

• This slight difference in shape affects how
  extreme a score you need to reject the null
  hypothesis.
• As always, to reject the null hypothesis, your
  sample mean has to be in an extreme section of
  the comparison distribution of means.

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The t Distribution

• However, if the distribution has more of its
  means in the tails than a normal curve would
  have, then the point where the rejection region
  begins has to be further out on the comparison
  distribution.
• Thus, it takes a slightly more extreme sample
  mean to get a significant result when using a t
  distribution than when using a normal curve.

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The t Distribution

• For example, using the normal curve, 1.96 is the
  cut-off for a two-tailed test at the .05 level of
  significance.
• On a t distribution with 3 degrees of freedom (a
  sample size of 4), the cutoff is 3.18 for a
  two-tailed test at the .05 level of significance.
• If your estimate is based on a larger sample of
  7, the cutoff is 2.45, a critical score closer to
  that for the normal curve.

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The t Distribution

• If your sample size is infinite, the t
  distribution is the same as the normal curve.

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• Since it takes into account the changing shape of
  the distribution as n increases, there is a
  separate curve for each sample size (or degrees
  of freedom).
• However, there is not enough space in the table
  to put all of the different probabilities
  corresponding to each possible t score.
• The t table lists commonly used critical regions
  (at popular alpha levels).

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• If your study has degrees of freedom that do not
  appear on the table, use the next smallest number
  of degrees of freedom.
• Just as in the normal curve table, the table
  makes no distinction between negative and
  positive values of t because the area falling
  above a given positive value of t is the same as
  the area falling below the same negative value.

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The t Test for a Single Sample Example

• You are a chicken farmer if only you had paid
  more attention in school. Anyhow, you think that
  a new type of organic feed may lead to plumper
  chickens. As every chicken farmer knows, a fat
  chicken sells for more than a thin chicken, so
  you are excited. You know that a chicken on
  standard feed weighs, on average, 3 pounds. You
  feed a sample of 25 chickens the organic feed for
  several weeks. The average weight of a chicken
  on the new feed is 3.49 pounds with a standard
  deviation of 0.90 pounds. Should you switch to
  the organic feed? Use the .05 level of
  significance.

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Hypothesis Testing

• State the research question.
• State the statistical hypothesis.
• Set decision rule.
• Calculate the test statistic.
• Decide if result is significant.
• Interpret result as it relates to your research
  question.

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The t Test for a Single Sample Example

• State the research question.
• Does organic feed lead to plumper chickens?
• State the statistical hypothesis.

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• Set decision rule.

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The t Test for a Single Sample Example

• Calculate the test statistic.

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The t Test for a Single Sample Example

• Decide if result is significant.
• Reject H0, 2.72 gt 1.711
• Interpret result as it relates to your research
  question.
• The chickens on the organic feed weigh
  significantly more than the chickens on the
  standard feed.

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The t Test for a Single Sample Try in pairs

• Odometers measure automobile mileage. How
  close to the truth is the number that is
  registered? Suppose 12 cars travel exactly 10
  miles (measured beforehand) and the following
  mileage figures were recorded by the odometers
• 9.8, 10.1, 10.3, 10.2, 9.9, 10.4, 10.0, 9.9,
  10.3, 10.0, 10.1, 10.2
• Using the .01 level of significance, determine
  if you can trust your odometer.
• s .19
• Mean 10.1

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Hypothesis Testing

• State the research question.
• State the statistical hypothesis.
• Set decision rule.
• Calculate the test statistic.
• Decide if result is significant.
• Interpret result as it relates to your research
  question.

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The t Test for a Single Sample Example

• State the research question.
• Are odometers accurate?
• State the statistical hypotheses.

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The t Test for a Single Sample Example

• Set the decision rule.

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The t Test for a Single Sample Example

• Calculate the test statistic.

X2 96.04 102.01 106.09 104.04 98.01 108.16 100.00
98.01 106.09 100.00 102.01 104.04 1224.50
X 9.8 10.1 10.3 10.2 9.9 10.4 10.0 9.9 10.3 10.0 1
0.1 10.2 121.20
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The t Test for a Single Sample Example

• Decide if result is significant.
• Fail to reject H0, 1.67lt3.106
• Interpret result as it relates to your research
  question.
• The mileage your odometer records is not
  significantly different from the actual mileage
  your car travels.

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Confidence Intervals

• You can estimate a population mean based on
  confidence intervals rather than statistical
  hypothesis tests.
• A confidence interval is an interval of a certain
  width, which we feel confident will contain the
  population mean.
• You are not determining whether the sample mean
  differs significantly from the population mean.
• Instead, you are estimating the population mean
  based on knowing the sample mean.

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When to Use Confidence Intervals

• If the primary concern is whether an effect is
  present, use a hypothesis test.
• You should consider using a confidence interval
  whenever a hypothesis test leads you to reject
  the null hypothesis, in order to determine the
  possible size of the effect.

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The t Test for a Single Sample Example

• You are a chicken farmer if only you had paid
  more attention in school. Anyhow, you think that
  a new type of organic feed may lead to plumper
  chickens. As every chicken farmer knows, a fat
  chicken sells for more than a thin chicken, so
  you are excited. You know that a chicken on
  standard feed weighs, on average, 3 pounds. You
  feed a sample of 25 chickens the organic feed for
  several weeks. The average weight of a chicken
  on the new feed is 3.49 pounds with a standard
  deviation of 0.90 pounds. Should you switch to
  the organic feed? Construct a 95 percent
  confidence interval for the population mean,
  based on the sample mean.

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The t Test for a Single Sample Example
Construct a 95 percent confidence interval.
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The t Test for a Single Sample Example
Construct a 99 percent confidence interval.
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Confidence Intervals

• Notice that the 99 percent confidence interval is
  wider than the corresponding 95 percent
  confidence interval.
• The larger the sample size, the smaller the
  standard error, and the narrower (more precise)
  the confidence interval will be.

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Confidence Intervals

• Its tempting to claim that once a particular 95
  percent confidence interval has been constructed,
  it includes the unknown population mean with a 95
  percent probability.
• However, any one particular confidence interval
  either does contain the population mean, or it
  does not.
• If a series of confidence intervals is
  constructed to estimate the same population mean,
  approximately 95 percent of these intervals
  should include the population mean.

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Next Week

• Finish Chps. 12 13
• You are now ready to ready to do the tutorial and
  the first problem set of homework 4