Stats week2

• Test Statistic


  A numerical value calculated from a sample that is used to determine whether to reject or fail to reject the null hypothesis
• Choosing the appropriate test statistic

  1. When the population variance is assumed to be known

  2. When the population variance is assumed to be unknown

  3. When the Central Limit Theorem is to be used
• 1. test


  A statistical test used when the population standard deviation is known
• 1. test


  A statistical test used when the population standard deviation is unknown
• When the population variance is known

  The appropriate test statistic is z-test
• When the population variance is unknown


  The appropriate test statistic is t-test
• When the Central Limit Theorem applies


  The appropriate test statistic is z-test or t-test depending on whether the population variance is known or unknown
• Null hypothesis (H0)


  A statement that there is no difference between a parameter (e.g. population mean) and a specific value (e.g. sample mean)
• Alternative hypothesis (Ha)


  A statement that there exists a difference between a parameter (e.g. population mean) and a specific value (e.g. sample mean)
• The most appropriate test statistic to use when the population standard deviation is known is z-test
• The most appropriate test statistic to use when the population standard deviation is unknown is t-test
• There are two statistical tests that are usually utilized for testing on a population mean: the z test and the t test
• z test


  Used when the standard deviation of the population σ is known
• t test


  Used when the standard deviation of the population σ is unknown
• The general formula for a test value or test statistic is: (observed value - expected value) / standard error
• z test statistic (zc)

  zc = (x̄ - μ) / (σ/√n)
• Conditions for using z test

  • n ≥ 30 and σ is known
  • Population is normally distributed for n < 30 and σ is known
• t test statistic (tc)


  tc = (x̄ - μ) / (s/√n)
• Conditions for using t test

  • Population is normally or approximately normally distributed, and σ is unknown
  • Sample size is small (n < 30)
• When n ≥ 30 and σ is unknown, the t test statistic tc = (x̄ - μ) / (s/√n) can be approximated by the z test statistic zc = (x̄ - μ) / (σ/√n)
• The t distribution approximates the z distribution as the sample size n goes to infinity
• Test statistics and when to use them

  • z test statistic (zc) when σ is known and n ≥ 30 or population is normally distributed for n < 30
  • t test statistic (tc) when σ is unknown and population is normally or approximately normally distributed, and n < 30
• z statistic


  Approximating the z statistic 𝒛𝒄 ≈

  𝒙̅−𝝁

  ��

  √��

  as n goes large
• The sampling distribution of the sample means which technically is a t distribution would look like/approximating a standard normal distribution (z distribution)
• As n goes infinitely large the t distribution is becoming the z distribution, having same values as the z distribution
• When n≥30 and 𝛔 𝐢𝐬 𝐮𝐧𝐤𝐧𝐨𝐰𝐧, the test statistic 𝒕𝒄 =

  𝒙̅−𝝁

  ��

  √��

  will be compared to a tabular value (z tabular value or z critical value) coming not from the t distribution but from the z distribution (standard normal distribution) to arrive at a decision
• Conditions/Assumptions


  • Population Standard Deviation σ is unknown
  • Sample size �� 𝟑𝟎 and population is normally distributed
  • Sample is a random sample
• Appropriate Test Statistic


  𝑡𝑐 = 𝑥̅ − 𝜇

  ��

  √��
• Statistical Test To Use

  t test
• Conditions/Assumptions


  • Population Standard Deviation σ is unknown
  • Sample size 𝒏 ≥ ��𝟎
  • Sample is a random sample
• Statistical Test To Use

  z test
• Although z test is commonly used for the given conditions, other statisticians are very conservative that they would opt to use t test
• Activity 1 Questions


  • What is the Null Hypothesis (Ho)
  • What is the Alternative Hypothesis (Ha)
  • Is this a 1 tailed test or two tailed test?
  • Do you think she made the right decision to use 𝑡𝑐 =

  𝑥̅−𝜇

  𝑠

  √𝑛

  as her test statistic? Why?

  • Do you think she made the right decision to use z test as the statistical test for the mean? Why?
  • What could be the possible consequences when we have not used the appropriate test statistic and/or statistical test?
  • If you are in the shoes of the medical researcher and you found out that you made a mistake, what would you do?
• What are test statistics?
• How do we identify appropriately the test statistic to be used in hypothesis testing?
• Why do you think it is important for us to identify appropriately the test statistic to be used in hypothesis testing?
• What might happen when we have used a t statistic as our test statistic when we are to use z statistic and vice versa? Do you think this will have serious implications?
• In this module, I have learned that
• What I Can Do Questions

  • What is the Null hypothesis?
  • What is the Alternative Hypothesis?
  • What test statistics should be used?
  • What type of statistical test must be employed?
• Assessment Questions


  • Which of the following is the correct test statistic to use?
  • What type of statistical test will best suit this research?
  • Which among the following statement/s is/are true?