Todays Goals

1
Todays Goals

• Perform a hypothesis test
• No more HW this semester
• Mini Project due May 11

2
Hypothesis Testing

• The result of a hypothesis test is either
• The null hypothesis is rejected
• This is a strong result. It indicates that your
  alternative hypothesis has convincing data behind
  it.
• The null hypothesis fails to be rejected
• This is a weak result.
• It DOES NOT imply that the null hypothesis is
  true.
• Only that there is not a convincing amount of
  data to support the alternative
• If you want to prove something, it should be your
  alternative hypothesis, not your null hypothesis.

3
Hypothesis Testing normal distribution with
known variance

• H0 mean m0
• Our test statistic is

4
Example

• The mean length of a part is expected to be 30mm.
  We are interested in determining whether for the
  month of March, the mean length differs from 30
  mm.
• Null Hypothesis ? Ho m 30mm
• Alternative Hypothesis ? Ha m ? 30mm
• This is a two-tailed test
• Designed to detect departures of a parameter from
  a specified value in both directions
• Assume that the population standard deviation
  s2mm
• A sample of size 36 finds the sample mean length
  to be 29 mm.
• Is this difference statistically significant?

5
Example

• m0 30mm
• n 36
• s2mm
• xbar 29
• Calculate
• The rejection region for 1 is -z.005 -2.575
• Since z -3 lt -2.575 the null hypothesis is
  rejected at the 1 level. It is very unlikely
  that the mean is actually 30 mm.

6
Example

• A cereal maker claims that their cereal boxes
  average 16 oz. of cereal per box. A consumer
  advocate claims that the average weight is in
  fact less than this.
• Assume that the weights vary normally with s0.5
  ozs
• The advocate measures the cereal in 5 randomly
  selected boxes and finds that the average weight
  is only 15.8 oz. Can he prove the cereal maker is
  cheating?
• Is the null hypothesis H0 mean 16 rejected in
  favor of the alternative hypothesis H1 mean lt
  16, at the 5 level?
• True if rejected
• False if fail to reject

7
z.05 1.645 What would the sample mean have to
have been in order to prove the company was
cheating at the 5 level?
8

• The null hypothesis is that mu 1.
• The alternative hypothesis is that mu gt 1
• We tested historical data and found that we
  failed to reject the null hypothesis at the 1
  level.
• True or False This proves that mu 1

9
Hypothesis Testing normal distribution with
known variance

• H0 mean m0
• Our test statistic is

10
Hypothesis Testing large sample

• H0 mean m0
• Our test statistic is

11
Hypothesis Testing Normal Population, unknown
variance

• H0 mean m0
• Our test statistic is

12
Example maximum weight of lift

• Assume MWL is normally distributed
• Does the data suggest the mean MWL exceeds 25 at
  a significance level of .05?
• Data 25.8, 36.6, 26.3, 21.8, 27.2
• xbar 27.54
• s 5.47
• True if the data suggests that it exceeds 25.
• False otherwise.

13
Example maximum weight of lift

• Assume MWL is normally distributed
• Does the data suggest the mean MWL exceeds 25 at
  a significance level of .05?
• Data 25.8, 36.6, 26.3, 21.8, 27.2
• xbar 27.54
• s 5.47
• H0 mean is 25 or less
• H1 mean is greater than 25

t.05,4 2.132 t is not greater than 2.132 Fail
to reject
14
Hypothesis Testing proportions, large sample
tests

• H0 p p0
• Our test statistic is

15
Example

• A manufacturer claims that his parts have a
  defect rate of 1.
• How can we set up a hypothesis test to see if it
  is a reasonable claim?

16
Example

• A manufacturer claims that his parts have a
  defect rate of 1.
• H0 p 1
• H1 p lt 1
• Test 1000 parts. Find 8 defects.
• Can the manufacturer support his claim?

17
Example

• A manufacturer claims that his parts have a
  defect rate of 1.
• H0 p 1
• H1 p lt 1
• Test 1000 parts. Find 8 defects.
• Can the manufacturer support his claim?

z.01 2.33 z.05 1.645
18
Example

• A manufacturer claims that his parts have a
  defect rate of 1.
• H0 p 1
• H1 p lt 1
• Test 1000 parts. Find 8 defects.
• Can the manufacturer support his claim?

z.01 2.33 z.05 1.645
Z is not in the rejection region. The claim is
not supported at the 1 or 5 level.