Hypothesis Testing (Decisions on Means, Proportions and Variances)

1
Hypothesis Testing(Decisions on Means,
Proportions and Variances)

• QSCI 381 Lecture 28
• (Larson and Farber, Sect 7.3 - 7.5)

2
Overview

• Yesterday, we dealt with the case in which the
  sample size was large. Today we address the
  issue of how to handle cases in which the sample
  size is small.
• By small we mean less than 30 samples note
  that the population needs to be normal (or at
  least approximately normal).

3
The Test Procedure

• State the claim mathematically and verbally.
  Identify the null and alternative hypotheses.
• H0 ? Ha ?
• Specify the level of significance.
• ??
• Identify the degrees of freedom (d.f n-1) and
  hence the critical value(s) and the rejection
  region.
• Determine the standardized test statistic
• Check whether t is in the rejection region.

4
Example-I(Comparison of Treatments)

• Under normal feeding conditions, a given fish is
  2 kg after two years of growth.
• We develop an alternative feeding program (which
  is cheaper).
• We feed a sample of 20 fish and, after 2 years,
  the fish are 1.9 kg on average (s.d. 0.4).
• Can someone reject the claim that fish fed on our
  diet are greater or equal to those fed using the
  usual diet.

5
Example-II
Note that We multiply by 2

1. H0 ??2 Ha ?lt2.
2. ?0.05.
3. d.f 19 tc-1.73 (TINV(20.05,20-1))
4. t (1.9-2)/(0.4/?20)-1.11
5. t is not in the rejection region (even though the
   sample mean is less than the value we are
   comparing it with).

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Example-III
Rejection region
Do not reject
What would have happened had our standard
deviation been 0.2 rather than 0.4? Just because
we cannot reject the null hypothesis does not
mean that it is not false!
7
Hypothesis Tests for Proportions-I

• Recall that we can approximate a binomial
  distribution, B(n,p) with the normal
  distribution, as long as np ? 5,
  nq ? 5.
• We will make use of this to define a hypothesis
  test for proportions.

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Hypothesis Tests for Proportions-II

• State the claim mathematically and verbally.
  Identify the null and alternative hypotheses.
• H0 ? Ha ?
• Specify the level of significance, ?, and hence
  the rejection region.
• Determine the standardized test statistic
• Check whether z is in the rejection region

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Example-I

• Under the Sustainable Fisheries Act, stocks
  should be managed so that they are at a target
  level. The expectation is therefore that 50 of
  stocks are above their target levels.
• We review the status of 300 stocks and find that
  47 (a slight exaggeration of reality) are below
  their target levels. Can we reject the claim that
  management is achieving its goal? Use a
  significance level of 0.01.

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Example-II

• H0 p??0.5 Ha pgt0.5.
• ?0.01. The rejection region is z gt 2.33.
• The standardized test statistic is
• We fail to reject H0.
• The U.S. government supplies statistics on its
  performance managing fisheries. See if
• its performance is really as good as this??

http//www.nmfs.noaa.gov/sfa/statusoffisheries/S
OSmain.htm
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Hypothesis Tests for Variances-I(the ?2 test)

• State the claim mathematically and verbally.
  Identify the null and alternative hypotheses.
• H0 ? Ha ?
• Specify the level of significance, ?, and
  determine the degrees of freedom (d.f. n-1)
• Find the rejection region (note that the
  chi-square distribution is not symmetric)
• Determine the standardized test statistic
• Check whether ?2 is in the rejection region.
• Note that the population must be normally
  distributed to use this test.

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Hypothesis Tests for Variances-II(Finding the
critical values and the rejection region)

• We need
• The level of significance.
• The degrees of freedom
• Whether this is a
• right-tailed test (?2 corresponding to ?)
• left-tailed test (?2 corresponding to 1-?)
• two-tailed test (?2 corresponding to ½? and 1- ½?)

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Hypothesis Tests for Variances-III(Finding the
critical values and the rejection region)

• Given n26 (i.e. d.f25). Find the rejection
  regions for ?0.1

CHIINV(0.9,25)
CHIINV(0.1,25)
CHIINV(0.05,25) CHIINV(0.95,25)
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Example-I

• A restaurant claims that the standard deviation
  of the length of its serving times is less than
  2.9 minutes. A random sample of 23 serving times
  has a standard deviation of 2.1 minutes. Is there
  enough evidence to support the claim at the 0.1
  level of significance?

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Example-II

1. The null hypothesis is ??2.9 (not ?lt2.9 why).
2. The level of significance is 0.1 and degrees of
   freedom is 22 (23-1).
3. This is left-tailed test so we determine the
   critical value of ?2 to be 14.042
   (CHIINV(0.9,22)).
4. We now calculate the standardized test statistic
5. Now 11.54 lt 14.042 so we reject H0.

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Caveats About Hypothesis Testing

• Check the assumptions of the test
• Random sampling
• Must the population be normal?
• Not rejecting the null hypothesis is not the same
  as proving it to be true. We should never say
  that the null hypothesis is proven if the p-value
  ? ?.
• Hypothesis tests focus on Type I error, but never
  forget about type II error.