Getting Started with Hypothesis Testing The Single Sample

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Transcript of Getting Started with Hypothesis Testing The Single Sample

  • Slide 1
  • Getting Started with Hypothesis Testing The Single Sample
  • Slide 2
  • Outline Remembering the binomial situation and z-score basics Hypothesis testing with the normal distribution When is unknown the t distribution One vs. Two-tails Problems
  • Slide 3
  • Recall the binomial We were able to do hypothesis testing regarding a proportion (of success) We created a probability distribution with respect to the expected probability of success, and then calculate the observed p-value for our specific result For example: H 0 : =.5 Probability if obtained 9/10 or 10/10 p = ~.01
  • Slide 4
  • Continuous measures If we know the population mean and standard deviation, or just want to speak about our sample, for any value of X we can compute a z-score Z-score tells us how far above or below the mean a value is in terms of standard deviation
  • Slide 5
  • Hypothesis testing using the normal If we were to test a hypothesis regarding our sample mean we must consult the sampling distribution and now are dealing with the standard error Our formula is the same as before, but substitutes our sample mean for an individual score and the standard error (regarding the sampling distribution) for the population standard deviation The tail probability is our observed p-value, and based on that we can decide whether our sample comes from a population suggested by the null hypothesis
  • Slide 6
  • Conceptual summary thus far H 0 : = some value Sample mean does not equal H 0 But how different is it? Is it what we would typically expect due to sampling variability or extreme enough to think that our sample does not come from such a population suggested by the null hypothesis?
  • Slide 7
  • Z to t In most situations we do not know However the sample standard deviation has properties that make it a good estimate of the population value We can use our sample standard deviation to estimate the population standard deviation However, if we use the normal distribution probabilities, they would be incorrect
  • Slide 8
  • t-test Which leads to: where And degrees of freedom (n-1)
  • Slide 9
  • Interpretation How many standard deviations away from the population mean is my sample mean in terms of the sampling distribution of means
  • Slide 10
  • Whats the difference? Why a t now not a z? The difference involves using our sample standard deviation to estimate the population standard deviation Standard deviation is positively skewed, and so slightly underestimates the population value As we have discussed it is actually a biased estimate Our standard error part of the formula will also be smaller than it should larger value of z than should be Increased type I error
  • Slide 11
  • Estimating Because we are trying to estimate , how well s does this depends on the sample size When n is larger, s is closer to When degrees of freedom = then t = z As N gets larger and larger the t distribution more closely approximates the normal distribution
  • Slide 12
  • Example The UNT Psychology Department claims in its recruiting literature that its graduate students get an average of 8 hours of sleep a night Collected sleep data from 25 grad students, this sample has a mean of 7.2 hours sleep, s = 1.5
  • Slide 13
  • Plug in the numbers Formula where t = (7.2 - 8)/(1.5/sqrt(25)) = -0.8/0.3 = -2.667 What else do we need to know?
  • Slide 14
  • Critical value of t One approach df = n-1 t.05 (24) = +2.064 1 The t obtained [-2.667] falls beyond the critical value Therefore p