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Getting Started with Hypothesis Testing The Single Sample
Outline • Remembering the binomial situation and z-score basics • Hypothesis testing with the normal distribution • When σ is unknown – the t distribution • One vs. Two-tails • Problems
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: H0: π = .5 Probability if obtained 9/10 or 10/10 p = ~.01 Recall the binomial
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
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 Hypothesis testing using the normal
Conceptual summary thus far • H0: μ = some value • Sample mean does not equal H0 • 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?
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
t-test • Which leads to: • where • And degrees of freedom (n-1)
Interpretation • How many standard deviations away from the population mean is my sample mean in terms of the sampling distribution of means
What’s 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
Estimating s • 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
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
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?
Critical value of t • One approach • df = n-1 • t.05(24) = +2.0641 • The t obtained [-2.667] falls beyond the critical value • Therefore p < .05 • Whose hypothesis testing approach is this? • Or better, go by specific p-value provided by statistical software • p = .0072 • Whose hypothesis testing approach is this? • Conclusion?
One vs. two-tailed probability • Note that just about every time you see a probability for zs, ts, and correlations it is a two-tailed probability • In other words, it’s the probability of that difference of that sizegreater or less than the null hypothesis value • This reflects complete ignorance about the research situation, which is rare • Unfortunately most test this way. Why? • Truly exploratory work • Poor estimation or ignorance of prior research • Habit .025 .025
One vs. two-tailed probability • A one-tailed test suggests that one expects a result of a certain type that you expect e.g. your result to be greater than the null hypothesis value • The informed situation is more statistically powerful, as here you can see the difference seen (and associated t-statistic) would not have to be as extreme to reach .05 probability .05
Your turn... • The average grade from year to year in undergrad statistics courses at UNT is an 81. This year the stats students (200 thus far) have an average of 83 w/ s = 10. Is this unusual? t.05(199)1= ? 2-tailed, i.e. probability associated with scores higher or lower than null. Before going in to this we would not have known whether they would be better or worse than the previous. t = ? p = ? Conclusion?
Problems with t • Wilcox and others note that when we sample from a non-normal population, assuming normality of the sampling distribution may be optimistic with small samples • Furthermore outliers have an influence on both the mean and sd used to calculate t • Has a larger effect on variance, increasing type II error due to std error increasing more so than the mean • This is not to say we throw the t-distribution out the window • If normality can be assumed, it is appropriate • However, if we cannot meet the normality assumption we may have to try a different approach • E.g. bootstrapping