Hypothesis tests for means

1
Section 8.2

• Hypothesis tests for means

2
Vocabulary

• Type I error Rejecting the null hypothesis and
  accepting the alternative hypothesis when the
  null hypothesis is true.
• Type II error. Failing to reject the null
  hypothesis when the alternative hypothesis is
  true.
• Significance level (a) - The preset probability
  of a type I error.

3
Steps of Hypothesis Test

• Give two alternative Claims or Hypotheses.
• The null hypothesis (H0) is what will be believed
  unless proven otherwise.
• The alternative hypothesis (Ha) is the hypothesis
  that will be accepted only if proven.
• Establish a Test Statistic.
• Agree on a significance level and determine the
  critical region accordingly.
• Perform the experiment and draw a conclusion.
• Reject null hypothesis or Fail to reject null
  hypothesis.

4
Tests about the mean .

• Case I. Suppose X is a N(m,s2) population where
  s is known.
• Test the hypotheses
• Take a sample of size n. Assuming the null
  hypothesis is true, what is the distribution of
  the sample mean?
• Give a test statistic that, assuming the null
  hypothesis is true, has a standard normal
  distribution.
• Give a critical region for the hypothesis test.

5
Three alternative hypotheses.Ha determines form
of critical region.

6
Example

• A certain car model is advertised to get 32 mpg.
  A random sample of 5 drivers and the gas
  mileage they got on their last tank of gas yields
  30 34 31 26 and 28 mpg. The sample mean is 29.8.
  Is this evidence that the average mpg is less
  than 32? (Assume the mpg is normally distributed
  with a standard deviation s2mpg.)
• Conduct hypothesis test (use a 0.05).
• Hypotheses
• Test Statistic
• Critical Region
• Decision

7
Further Questions on Example

• How many standard deviations(of the sample mean)
  below the null value is the observed sample mean
  of ?
• What would be the critical region for a0.01?
  What would be the decision in this case?
• Given the observed sample mean of, what is the
  smallest a at which the null hypothesis can be
  rejected (based on n5)?

8
Determining Type II error.

• Let m? denote a particular value of the
  alternative hypothesis. b(m?) is the probability
  of failing to reject the null when the true mean
  is m?. This is a Type II error.
• In the previous example (a0.05), calculate
  b(31).
• Calculate b(30).

9
Generalities for type II error in Case I.

• For a lower tailed test, the general formula is
• The general formula depends on the alternative
  hypothesis. See book page 325.
• The further m? is from the null hypothesis, the
  easier it is to detect the difference and the
  smaller the type II error is.

10
Sample Size Calculations

• Power is 1-b. This is the probability that an
  alternative value of the mean will be detected.
• In the previous example what sample size is
  needed to detect a true mean of 31 with a power
  of 0.95?

11
General formulas for sample size determination in
Case I.

• When doing a one-tailed test (either lower as in
  example or upper) the formula for sample size is
• The general from for a two sided test

12
Summary Case I.

• Use z critical values.
• Depending on form of alternative hypothesis.
• Find rejection region for a given a.
• Probability of type II error for a given value of
  alternative.
• Sample size calculation for a given a, b, and
  value of the alternative.

13
Tests about the mean .

• Case II. Suppose X is a population with mean m.
  Suppose the sample size is large. The rule of
  thumb is larger than 40.
• Test the hypotheses
• Take a sample of size n. Assuming the null
  hypothesis is true, what is the distribution of
  the sample mean?
• Then the following statistic approximately
  follows a standard normal distribution?
• Give a critical region for the hypothesis test.

14
Three alternative hypotheses.Ha determines form
of critical region.

15
Type II error and Sample Size

• By using a plausible value for s instead of s,
  these calculations can be done as in Case I.
• If s is slightly too large, the calculated bs
  and n will be larger than actual.
• Otherwise, calculations can be done as in Case
  III to be shown.

16
Summary Case II.

• Use z critical values.
• Depending on form of alternative hypothesis.
• Find rejection region for a given a.
• the following should be done as in case I
  (estimate s) or case III
• Probability of type II error for a given value of
  alternative.
• Sample size calculation for a given a, b, and
  value of the alternative.

17
Tests about the mean .

• Case III. Suppose X is a N(m,s2) population
  where s is not known.
• Test The hypotheses
• Take a sample of size n. Assuming the null
  hypothesis is true, what is the distribution of
  the sample mean?
• Give a test statistic that, assuming the null
  hypothesis is true, has a t distribution.
• What are the degrees of freedom?
• Give a critical region for the hypothesis test.

18
Three alternative hypotheses.Ha determines form
of critical region.

19
Example

• A certain car model is advertised to get 32 mpg.
  A random sample of 5 drivers and the gas
  mileage they got on their last tank of gas yields
  30 34 31 26 and 28 mpg. (x_bar 29.8, s3.03)
  Assuming that mpg is normally distributed, is
  this evidence that the average mpg is less than
  32?
• Conduct hypothesis test (use a 0.05).
• Hypotheses
• Test Statistic
• Critical Region
• Decision

20
Type II Error and Sample Size

• The general ideas behind type II error and sample
  size calculations are the same as before, but the
  calculations are more tedious.
• Calculating b(31) involves
• Use Minitab.

21
Summary Case III.

• Use t critical values.
• Depending on form of alternative hypothesis.
• Find rejection region for a given a.
• Using Minitab
• Probability of type II error for a given value of
  alternative.
• Sample size calculation for a given a, b, and
  value of the alternative.
• (Book gives instructions for using table in book,
  but not required to learn this.)

22
Summary

• Recognize correct form of alternative.
• Recognize which of the three cases.
• For each of the three cases
• Find rejection region.
• Calculate probability of type II error.
• Calculate Sample Size for a give a, b, and value
  of the alternative.