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Hypothesis Testing

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  • 1. Normal Distribution and Hypothesis Testing STR1K
  • 2.  
  • 3. Characteristics
    • Bell-shaped , depends on standard deviation
    • Continuous distribution
    • Unimodal
    • Symmetric about the vertical axis through the mean μ
    • Approaches the horizontal axis asymptotically
    • Total area under the curve and above the horizontal is 1
  • 4. Characteristics
    • Approximately 68% of observations fall within 1 σ from the mean
    • Approximately 95% of observations fall within 2 σ from the mean
    • Approximately 99.7% of observations fall within 3 σ from the mean
  • 5. 68.27% 95.45 % 99.73%
  • 6. Standard Normal Distribution
    • Special type of normal distribution where μ =0
    • Used to avoid integral calculus to find the area under the curve
    • Standardizes raw data
    • Dimensionless Z-score
    Z = X - μ 0 σ
  • 7.
    • Example 1
    • Given the normal distribution with μ = 49 and σ = 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 μ = 2.5; σ = 1.18 n = 3 μ xbar = 2.5
  • 10. Central Limit Theorem
    • Given a distribution with a mean μ and variance σ² , the sampling distribution of the mean approaches a normal distribution  with a mean (μ) and a variance σ²/N as N, the sample size, increases. 
  • 11. Characteristics
    • The mean of the population and the mean of the sampling distribution of means will always have the same value . 
    • The sampling distribution of the mean will be normal  regardless of the shape of the population distribution.
  • 12. N(70, 16)
  • 13. N(70,1)
  • 14. N(70,.25)
  • 15. Characteristics
    • As the sample size increases , the distribution of the sample average becomes less and less variable.
    • Hence the sample average X bar approaches the value of the population mean  μ . 
  • 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 a conjecture or assertion about a parameter
    • Null v. Alternative hypothesis
      • Proof by contradiction
      • Null hypothesis is the hypothesis being tested
      • Alternative hypothesis is the operational statement of the experiment that is believed to be true
  • 19. One-tailed test
      • Alternative hypothesis specifies a one-directional difference for parameter
      • H 0 : μ = 10 v. H a : μ < 10
      • H 0 : μ = 10 v. H a : μ > 10
      • H 0 : μ 1 - μ 2 = 0 v. H a : μ 1 - μ 2 > 0
      • H 0 : μ 1 - μ 2 = 0 v. H a : μ 1 - μ 2 < 0
  • 20. Two-tailed test
      • Alternative hypothesis does not specify a directional difference for the parameter of interest
      • H 0 : μ = 10 v. H a : μ ≠ 10
      • H 0 : μ 1 - μ 2 = 0 v. H a : μ 1 - μ 2 ≠ 0
  • 21. Critical Region
    • Also known as the “rejection region”
    • Critical region contains values of the test statistic for which the null hypothesis will be rejected
    • Acceptance and rejection regions are separated by the critical value, Z .
  • 22. Type I error
    • Error made by rejecting the null hypothesis when it is true .
    • False positive
    • Denoted by the level of significance, α
    • Level of significance suggests the highest probability of committing a type I error
  • 23. Type II error
    • Error made by not rejecting (accepting) the null hypothesis when it is false .
    • False negative
    • Probability denoted by β
  • 24.  
  • 25. Notes on errors
    • Type I ( α ) and type II errors ( β ) are related . A decrease in the probability of one, increases the probability in the other.
    • As α increases , the size of the critical region also increases
    • Consequently, if H 0 is rejected at a low α , H 0 will also be rejected at a higher α .
  • 26.  
  • 27. Testing a Hypothesis on the Population Mean Z = X - μ 0 σ /√n t = X - μ 0 S /√n υ = n - 1 H 0 Test Statistic H a Critical Region σ known μ = μ 0 μ < μ 0 μ > μ 0 μ ≠ μ 0 z < -z α z > z α |z| > z α /2 σ unknown μ = μ 0 μ < μ 0 μ > μ 0 μ ≠ μ 0 t < -t α t > t α |t| > t α /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.