7.1 Stuff

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7.1 Stuff

• Major applications of inferential statistics
• Estimate the value of a population parameter
• Test some claim (or hypothesis) about a population

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7.1 Stuff

• A statistical hypothesis is a claim or statement
  about a population parameter.
• A hypothesis test is a process that uses sample
  statistics to test a claim about the value of a
  population parameter.

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7.1 Examples

• A university publicizes that the proportion of
  its students who graduate in 4 years is 82.
• A water faucet manufacturer announces that the
  mean flow rate of a certain type of faucet is
  less than 2.5 gallons per minute.
• A cereal company advertises that the mean weight
  of the contents of its 20-ounce size cereal boxes
  is more than 20 ounces.

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7.1 Stuff

• A null hypothesis H0 is a statistical hypothesis
  that contains a statement of equality, such as
  .
• The alternative hypothesis Ha is the complement
  of the null hypothesis. It is a statement that
  must be true if H0 is false and it contains a
  statement of strict inequality, such as
  .

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7.1 Examples

1. A university publicizes that the proportion of
   its students who graduate in 4 years is 82.
2. A water faucet manufacturer announces that the
   mean flow rate of a certain type of faucet is
   less than 2.5 gallons per minute.
3. A cereal company advertises that the mean weight
   of the contents of its 20-ounce size cereal boxes
   is more than 20 ounces.

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7.1 Stuff
Actual Truth of H0 Actual Truth of H0
Decision H0 is true H0 is false
Do not reject H0 Correct Decision Type II Error
Reject H0 Type I Error Correct Decision

• A Type I error occurs if the null hypothesis is
  rejected when it is true.
• A Type II error occurs if the null hypothesis is
  not rejected when it is false.

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7.1 Stuff

• In a hypothesis test, the level of significance
  is the maximum allowable probability of making a
  type I error. It is denoted by , the
  lowercase Greek letter alpha.
• The probability of a type II error is denoted by
  , the lowercase Greek letter beta.

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7.1 Stuff
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7.1 Stuff

• If the null hypothesis is true, a P-value (or
  probability value) of a hypothesis test is the
  probability of obtaining a sample statistic with
  a value as extreme or more extreme than the one
  determined from the sample data.

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7.1 Example

• The proportion of drivers who admit to running
  red lights is greater than 0.5.

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7.1 Example

• The proportion of drivers who admit to running
  red lights is greater than 0.5.
• A survey of 880 randomly selected drivers showed
  that 56 of those respondents admitted to running
  red lights.

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7.1 Stuff

• Decision Rule based on P-value
• If , then reject H0.
• If , then fail to reject H0.

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7.1 Example

• The proportion of drivers who admit to running
  red lights is greater than 0.5.
• A survey of 880 randomly selected drivers showed
  that 56 of those respondents admitted to running
  red lights.
• At the 5 level of significance, there is enough
  evidence to support the claim that the proportion
  of drivers who admit to running red lights is
  greater than 0.5.