HYPOTHESIS TESTING

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HYPOTHESIS TESTING

• CHAPTER 3
• BCT 2053 APPLIED STATISTICS
• by
• SITI ZANARIAH SATARI
• FIST/FSKKP UMP 2009

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CONTENT

• 3.1 Introduction to Hypothesis Testing
• 3.2 Hypothesis Testing for Mean with known and
  unknown
• Variance
• 3.3 Hypothesis Testing for Difference Means with
  known
• and unknown Population Variance
• 3.4 Hypothesis Testing for Proportion
• 3.5 Hypothesis Testing for the Difference between
  Two
• Proportions
• 3.6 Hypothesis Testing for Variances and Standard
  
• Deviations
• 3.7 Hypothesis Testing for Two Variances and
  Standard
• Deviations

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3.1 Introduction to Hypothesis Testing

• OBJECTIVES After completing this chapter, you
  should be able to
• Describe the meaning of terms used in hypothesis
  testing.
• State the Null and Alternative Hypothesis

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General Terms in Hypothesis Testing

• Hypothesis
• A statement that something TRUE
• Statistical Hypothesis
• A statement about the parameters of one or more
  populations.
• Null Hypothesis (Ho)
• A hypothesis to be tested
• Alternative Hypothesis (H1)
• A hypothesis to be considered as an alternative
  to the null hypothesis

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How to make Null hypothesis

• Should have an equal sign
• Generally,

parameter
A value
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How to make Alternative hypothesis

• Should reflect the purpose of the hypothesis test
  and different from the null hypothesis
• Generally (3 types)

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Basic logic of Hypothesis Testing

• Accept null hypothesis
• if the sample data are consistent with the null
  hypothesis
• Reject null hypothesis
• if the sample data are inconsistent with the null
  hypothesis, so accept Alternative hypothesis
• Test statistics
• the statistics used as a basis for deciding
  whether the null hypothesis should be rejected
  (z, t, Khi, F)

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Basic logic of Hypothesis Testing

• Rejection (Critical) Region, (a)
• the set of values for the test statistics that
  leads to rejection of the null hypothesis
• Nonrejection (NonCritical) Region,(1 a)
• the set of values for the test statistics that
  leads to nonrejection of the null hypothesis
• Critical values
• the values of the test statistics that separate
  the rejection and nonrejection regions.
• A critical value is considered part of the
  rejection region
• In general, Reject Ho if test statistics gt
  critical value

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Rejection Region
Reject Ho
Reject Ho
Accept Ho 1 a
Critical value
Critical value
Reject Ho
Accept Ho 1 a
Critical value
Reject Ho
Accept Ho 1 a
Critical value
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Type I and II Error

• Type I Error
• - Rejecting the null hypothesis when it is in
  fact true

• Type II Error
• -Not rejecting the null hypothesis when it is in
  fact false

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Hypothesis Testing Common Phrase
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Example

• State the null and alternative hypotheses for
  each conjecture.
• A researcher thinks that if expectant mothers use
  vitamin pills, the birth weight of the babies
  will increase. The average birth weight of the
  population is 8.6 pounds.
• An engineer hypothesizes that the mean number of
  defects can be decreased in a manufacturing
  process of compact disks by using robots instead
  of humans for certain tasks. The mean number of
  defective disks per 1000 is 18.
• A psychologist feels that playing soft music
  during a test will change the results of the
  test. The psychologist is not sure whether the
  grades will be higher or lower. In the past, the
  mean of the scores was 73.

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Steps In Hypothesis Testing

• Define the parameter used
• Define the null and alternative hypothesis
• Define all the given information
• Chose appropriate Test Statistics (z,t,chi,F)
• Find Critical value
• Test the hypothesis (rejection region)
• Make conclusion there is enough evidence to
  reject/accept the claim at a

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3.2 Hypothesis Testing for Mean with
known and unknown Variance

• OBJECTIVES After completing this chapter, you
  should be able to
• Test mean when s ² is known.
• Test mean when s ² is unknown.

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Hypothesis Testing for Mean µ
Where
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Example 1 Hypothesis testing for mean µ with
known s²

• A lecturer state that the IQ score for IPT
  students must be more higher than other people
  IQs which is known to be normally distributed
  with mean 110 and standard deviation 10. To prove
  his hypothesis, 25 IPT students were chosen and
  were given an IQ test. The result shows that the
  mean IQ score for 25 IPT students is 114. Can we
  accept his hypothesis at significance level, a
  0.05?

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Example 2 Hypothesis testing for mean µ with
unknown s² and n 30

• UMP students said that they have no enough time
  to sleep. A sample of 36 students give that the
  mean of sleep time is 6 hours and the standard
  deviation is 0.9 hours. It is known that the mean
  sleep time for adult is 6.5 hours. Can we accept
  their hypothesis at significance level, a 0.01?

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Example 3 Hypothesis testing for mean µ with
unknown s² and n lt 30

• In a wood cutting process to produce rulers, the
  mean of rulers height is set to be equal 100 cm
  at all times. If the mean height of rulers is not
  equal to 100 cm, the process will stop
  immediately. The height for a sample of 10 rulers
  produces by the process shows below
• 100.13 100.11 100.02 99.99 99.98
• 100.14 100.03 100.10 99.97 100.21
• Can we stop the process at significance level, a
  0.05?

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3.3 Hypothesis Testing for Difference Means with
known and unknown Population Variance

• OBJECTIVES After completing this chapter, you
  should be able to
• Test the difference between two means when s s
  are known.
• Test the difference between two means when s s
  are unknown and equal.
• Test the difference between two means when s s
  are unknown and not equal.

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Hypothesis Testing for Different Mean with known
and unknown Variance
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Example 4 Hypothesis testing for µ1 µ2 with
known s1² and s2²

• The mean lifetime for 30 battery type A is 5.3
  hours while the mean lifetime for 35 battery type
  B is 4.8 hours. If the lifetime standard
  deviation for the battery type A is 1 and the
  lifetime standard deviation for the battery type
  B is 0.7 hours, can we conclude that the lifetime
  for both batteries type A and type B are same at
  significance level, a 0.05?

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Example 5 Hypothesis testing for µ1 µ2 with
unknown s1² s2², s1² ? s2², n1 30 n2 30

• The mean price of 30 acre of land in Cerok before
  a highway is build is RM20,000 per acre with
  standard deviation RM 3000 per acre. The mean
  price of 36 acre of land in Cerok after the
  highway is build is RM50,000 per acre with
  standard deviation RM 4000 per acre. Test a
  hypothesis that the new highway will increases
  the land price in Cerok among RM35,000 at
  significance level, a 0.05. Assume that the
  variances of land price in Cerok are not same
  before and after the highway is build.

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Example 6 Hypothesis testing for µ1 µ2 with
unknown s1² s2², s1² ? s2², n1 lt 30 n2 lt 30

• The mean lifetime for 10 battery type A is 5.3
  hours with standard deviation 1 hour while the
  mean lifetime for 8 battery type B is 4.8 hours
  with standard deviation 0.7 hours. Can we
  conclude that the lifetime for both battery type
  A and type B are same at significance level, a
  0.05? Assume that the variance lifetime for both
  batteries type A and type B are different.

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Example 7 Hypothesis testing for µ1 µ2 with
unknown s1² s2², s1² s2² , n1 30 n2 30

• Many studies have been conducted to test the
  effects of marijuana use on mental abilities. In
  a study, groups of light and heavy users of
  marijuana were tested for memory recall, with the
  result below.
• Item sort correctly by light marijuana users
• Item sort correctly by heavy marijuana users
• Use 0.01 significance level to test the claim
  that the population of heavy marijuana users has
  a lower mean than the light users if the variance
  population for both users are same.

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Example 8 Hypothesis testing for µ1 µ2 with
unknown s1² s2², s1² s2² , n1 lt 30 n2 lt 30

• Two catalyst are being analyzed to determine the
  mean yield of a chemical process. A test is run
  in the pilot plant and results are shown below.
• catalyst 1 91.50 94.18 92.18 95.39
  91.79 89.07 94.72 89.21
• catalyst 2 89.19 90.95 90.46 93.21
  97.19 97.04 91.07 92.75
• Is there any different between the mean
  yield? Use a 0.05 and assume the variances
  population are equal.

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3.4 Hypothesis Testing for Proportion

• OBJECTIVES After completing this chapter, you
  should be able to
• Test proportions using z-test.

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Hypothesis testing for proportion p
Where
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Example 9 Hypothesis testing
for proportion p

• A telephone company representative estimates that
  40 of its customers have call-waiting service.
  To test this hypothesis, she selected a sample of
  100 customers and found that 37 had call
  waiting. At a 0.05, is there enough evidence to
  reject the claim?

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Example 10 Hypothesis testing
for proportion p

• A group of scientist believes that their new
  medicine can heal 40 of patients. The current
  medicine in market can only heal 30 of patients.
  A research is done to test the hypothesis made by
  the scientists. The new medicine is given to the
  100 patients and it shows that only 26 patients
  are recovered. Can we accept their hypothesis at
  significance level a 0.05?

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3.5 Hypothesis Testing for the Difference between
two Proportions

• OBJECTIVES After completing this chapter, you
  should be able to
• Test the difference between two proportions.

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Hypothesis testing for the difference between two
proportions p1 p2
Where
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Example 11 Hypothesis testing for
difference proportion p1 p2

• In a sample of 200 surgeons, 15 thought the
  government should control health care. In a
  sample of 200 practitioners, 21 felt the same
  way. At a 0.01, is there a difference in the
  proportions between surgeons and practitioners?

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Example 12 Hypothesis testing for
difference proportion p1 p2

• Random samples of 747 Malaysian men and 434
  Malaysian women were taken. Of those sampled, 276
  men and 195 women said that they sometimes
  ordered dish without meat or fish when they eat
  out. Do the data provide sufficient evidence to
  conclude that, in Malaysia, the percentage of men
  who sometimes order a dish without meat or fish
  is smaller than the percentage of women who
  sometimes order a dish without meat or fish at
  significance level a 0.05?

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3.6 Hypothesis Testing for Variances and Standard
Deviations

• OBJECTIVES After completing this chapter, you
  should be able to
• Test single variance and standard deviation

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Hypothesis testing for variance s²
Where
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Example 13 Hypothesis Testing for s²

• In a wood cutting process to produce rulers, the
  variance of rulers height is set to be equal 2
  cm² at all times. If the variance of rulers
  height is not equal to 2 cm², the process will
  stop immediately. The height for a sample of 10
  rulers produces by the process shows below
• 100.23 100.11 100.42 99.66 99.68
• 100.14 100.33 100.10 99.50 100.21
• Can we stop the process at significance level a
  0.05?

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Example 14 Hypothesis Testing for s²

• A hospital administrator believes that the
  standard deviation of the number of people using
  outpatient surgery per day is greater than 8. A
  random sample of 15 days is selected and the
  standard deviation is 11.2. At a 0.05, is there
  enough evidence to support the administrators
  claim?

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3.7 Hypothesis Testing for Two Variances and
Standard Deviations

• OBJECTIVES After completing this chapter, you
  should be able to
• Test the difference between two variances.

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Hypothesis testing for variance ratio s1²/ s2²
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Example 15 Hypothesis Testing for difference
proportions s1²/ s2²

• Before service, a machine can packed 10 packets
  of sugar with variance weight 64 g² while after
  service the variance weight for 5 packets of
  sugar are 25 g². Do the services improve the
  packaging process at significance level, a 0.05?

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Example 16 Hypothesis Testing for difference
proportions s1²/ s2²

• A medical researcher whishes to see whether the
  variance of the heart rates (in beats per minute)
  of smokers is different from the variance of
  heart rates of people do not smoke. Two samples
  are selected, and the data are as shown below.
  Using a 0.01, is there enough evidence to
  support the claim?
• Smokers Nonsmokers

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Summary

• 0.01, 0.05 and 0.1 significance levels are
  usually used in testing a hypothesis.
• Hypothesis test are closely related to confidence
  interval. Whenever a confidence interval can be
  computed, a hypothesis test can also be
  performed, and vice versa.
• The End

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