INTRODUCTION TO HYPOTHESIS TESTING

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INTRODUCTION TO HYPOTHESIS TESTING
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PURPOSE

• A hypothesis test allows us to draw conclusions
  or make decisions regarding population data from
  sample data.

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The Logic of Hypothesis Tests

• Assume a population distribution with a specified
  population mean.
• State the hypothesized population mean (this
  statement is referred to as the null hypothesis).
• Draw a random sample from the population and
  calculate the sample mean.
• Determine the relative position on the
  calculated mean on the distribution of sample
  means. If the sample mean is close to the
  specified population mean, we do not have
  evidence to reject the hypothesized population
  mean.
• If the calculated sample mean is not close to
  the specified population mean, we conclude that
  our sample could not have been drawn from the
  hypothesized distribution, and thus, we
• reject the null hypothesis.

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Example

• The president of City Real Estate claims that
  the mean selling time of a home is 40 days after
  it is listed. A sample of 50 recently sold homes
  shows a sample mean of 45 days with a standard
  deviation of 20 days. Is the president correct?

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ONE SAMPLE HYPOTHESIS TESTS

• Applied to determine if the population mean is
  consistent with a specified value or standard
• Two tests
• the z- test
• the t-test

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ONE SAMPLE HYPOTHESIS TEST Large Sample

• Sample size ngt25
• Null and Alternative Hypothesis
• Ho m
• Ha m / or m gt or m lt

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ASSUMPTIONS z-TEST

• the underlying distribution is normal or the
  Central Limit Theorem can be assumed to hold
• the sample has been randomly selected
• the population standard deviation is known or the
  sample size is at least 25.

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Example

• A manufacturer of electric ovens purchases
  components with a specified heat resistance of
  8000. A sample of 36 components selected from a
  large shipment shows an average heat resistance
  of 7900 and a standard deviation of 200. Can the
  manufacturer conclude that the heat resistance of
  the glass components is less than 8000?

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ONE SAMPLE HYPOTHESIS TESTSmall Samples

• Null and Alternative Hypothesis
• Ho m
• Ha m / or m gt or m lt

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ASSUMPTIONS t-TEST

• The underlying distribution is normal or the CLT
  can be assumed to hold
• The samples have been randomly and independently
  selected from two populations
• The variability of the measurements in the two
  populations is the same and can be measured by a
  common variance.
• (There is a t-test that does not make this
  assumption it is available when using Minitab.)

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EXAMPLE

• A manufacturer uses a bottling process and will
  lose money if the bottles do not contain the
  labeled amount. Suppose a cola company labels the
  bottles as 20 oz. A sample of 16 bottles results
  in 19.6 oz and a standard deviation of 0.3 oz.
  Does the process need an adjustment?

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Paired Samples Test

• Find the difference in the paired values
• Treat the difference scores as one sample.
• Apply a one sample test.

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EXAMPLE

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TWO-SAMPLES HYPOTHESIS TESTS

• Applied to compare the values of two population
  means.

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The Distribution of the Difference Between Two
Independent Samples
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HYPOTHESIS TEST TWO INDEPENDENT SAMPLESLarge
Samples

• Sample Size n lt 25
• Null and Alternative Hypothesis
• Ho m1 m 2
• Ha m 1/ m 2 or m 1 gt m 2 or m 1 lt m 2

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HYPOTHESIS TEST TWO INDEPENDENT SAMPLESSmall
Samples

• Null and Alternative Hypothesis
• Ho m1 m2
• Ha m1 / m2 or m1 gt m2 or m1lt m 2

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Example

• Two machines are used in the manufacturer of
  steel rings. The quality control director wishes
  to know if she should conclude machine A is
  producing rings with a different inside diameter
  than those produced by machine B.

Type A Type B
N 40 40
Mean 2 1.5
Variance 0.001 0.002
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Example/Proportion

• Sports car owners complain that their cars are
  judged differently from sedans at the vehicle
  inspection station. Previous records indicate
  that 30 of all cars fail inspection on the first
  time. A random sample of 150 sports cars produced
  60 that failed. Is there a different standard?

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Estimating the Difference in Population Means

• For large samples, point estimates and their
  margin of error as well as confidence intervals
  are based on the standard normal (z)
  distribution.

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Example/Proportions

• In producing a particular component, the Shelby
  Co. has a defective rate of 2. In a sample of
  500, a contractor found a rate of 1. Has the
  quality improved?