Hypothesis Testing

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Transcript of Hypothesis Testing

  • 1. Normal Distribution and Hypothesis TestingSTR1K

2. 3. Characteristics

  • Bell-shaped , depends on standard deviation
  • Continuousdistribution
  • Unimodal
  • Symmetricabout the vertical axis through the mean
  • Approaches the horizontalaxis asymptotically
  • Total areaunder the curve and above the horizontalis 1

4. Characteristics

  • Approximately 68%of observations fall within1 from the mean
  • Approximately 95%of observations fall within2 from the mean
  • Approximately 99.7%of observations fall within3 from the mean

5. 68.27%95.45% 99.73% 6. Standard Normal Distribution

  • Special type ofnormal distribution where=0
  • Used to avoid integral calculus to find the area under the curve
  • Standardizesraw data
  • DimensionlessZ-score

Z =X - 0 7.

  • Example 1
  • Given the normal distribution with= 49 and= 8, find the probability that X assumes a value:
  • Less than 45
  • More than 50

8. Example 2 The achievement sores for a college entrance examination are normally distributed with the mean 75 and standard deviation equal to 10. What fraction of the scores would one expect to lie between 70 and 90. 9. Sampling Distribution

  • Distribution of all possible sample statistics

Population All Possible Samples Sample Means 1, 2, 3, 4 1, 2, 3 2.00 1, 2, 4 2.33 3, 4, 1 2.67 2, 3, 4 3.00 = 2.5; = 1.18 n = 3 xbar= 2.5 10. Central Limit Theorem

  • Given a distribution with amean andvariance , the samplingdistribution of themeanapproaches anormal distribution with a mean () and a variance /N as N, thesample size,increases.

11. Characteristics

  • Themean of the populationand themean of the sampling distributionof means will always have thesame value .
  • Thesampling distribution of the mean will be normal regardlessof the shape of the population distribution.

12. N(70, 16) 13. N(70,1) 14. N(70,.25) 15. Characteristics

  • As thesample size increases , thedistributionof the sample averagebecomes less and less variable.
  • Hence thesample average X barapproachesthe value of thepopulation mean .

16. Example 3 An electrical firm manufactures light bulbs that have a length of life normally distributed with mean and standard deviation equal to 500 and 50 hours respectively. Find the probability that a random sample of 15 bulbs will have an average life ofless than 475 hours. 17. HYPOTHESIS TESTING

  • Normal Distribution and Hypothesis Testing

18. Hypothesis Testing

  • A hypothesis is aconjecture or assertion about a parameter
  • Null v. Alternative hypothesis
    • Proof by contradiction
    • Null hypothesis is thehypothesis being tested
    • Alternative hypothesis is theoperational statement of the experimentthat is believed to be true

19. One-tailed test

    • Alternative hypothesisspecifies a one-directional differencefor parameter
    • H 0 := 10 v. H a :< 10
    • H 0 := 10 v. H a :> 10
    • H 0 : 1- 2= 0 v. H a : 1- 2> 0
    • H 0 : 1- 2= 0 v. H a : 1- 2< 0

20. Two-tailed test

    • Alternative hypothesisdoes not specify a directional differencefor the parameter of interest
    • H 0 := 10 v. H a : 10
    • H 0 : 1- 2= 0 v. H a : 1- 2 0

21. Critical Region

  • Also known as therejection region
  • Critical region contains values of the test statistic for which thenull hypothesis will be rejected
  • Acceptance and rejection regions are separated by thecritical value, Z .

22. Type I error

  • Error made byrejecting the null hypothesis when it is true .
  • False positive
  • Denoted by thelevel of significance,
  • Level of significance suggests the highest probability of committing a type I error

23. Type II error

  • Error made bynot rejecting (accepting) the null hypothesis when it is false .
  • False negative
  • Probability denoted by

24. 25. Notes on errors

  • Type I ( ) and type II errors ( ) are related . A decrease in the probability of one, increases the probability in the other.
  • Asincreases , the size of the critical region also increases
  • Consequently, ifH 0is rejected at a low , H 0willalso be rejected at a higher .

26. 27. Testing a Hypothesis on the Population Mean Z =X - 0 /n t=X - 0 S /n =n - 1 H 0 Test Statistic H a Critical Region known = 0 < 0 > 0 0 z < -z z > z |z| > z /2 unknown = 0 < 0 > 0 0 t < -t t > t |t| > t /2 28. critical value test statistic Reject H 0 29. critical value test statistic Do not reject H 0 30. Example 4 It is claimed that an automobile is driven on the average of less than 25,000 km per year.To test this claim, a random sample of 100 automobile owners are asked to keep a record of the kilometers they travel. Would you agree with this claim if the random sample showed an average of 23,500 km and a standard deviation of 3,900 km? Use 0.01 level of significance.