Statistical Tests P Values

The Impact of Sampling We are sampling We don’t expect every sample to look exactly like the population. There is going to be variability because of chance

Simulation – 20 trials Simulation – 10000 trials 1 2 3 4 5 6 7 8 9 10 Number Heads 1 2 3 4 5 6 7 8 9 10 Total Times occurred Probability of Occurring 0.0 0.05  0.15 0.15 0.25 0.2 0.1   0 Simulation – 10000 trials Number Heads 1 2 3 4 5 6 7 8 9 10 Total Times occurred 92 461 1154 1981 2537 2063 1117 479 101 Probability of Occurring 0.0005 0.009 0.046  0.115 0.198 0.245 0.206 0.117 0.048 0.010  0.0006

The Big Question… My simulation – 10000 trials What is the likelihood (probability) of having AT LEAST 8 heads in our sample (getting 8, 9, or 10 heads)? My simulation – 10000 trials Number Heads 1 2 3 4 5 6 7 8 9 10 Total Times occurred 92 461 1154 1981 2537 2063 1117 479 101 Probability of Occurring 0.0005 0.009 0.046  0.115 0.198 0.245 0.206 0.117 0.048 0.010  0.0006

The Big Question… My simulation – 10000 trials What is the likelihood (probability) of having AT LEAST 8 heads in our sample? p= 0.048 +0.01 + 0.006 so p=0.0586 (8) (9) (10) My simulation – 10000 trials Number Heads 1 2 3 4 5 6 7 8 9 10 Total Times occurred 92 461 1154 1981 2537 2063 1117 479 101 Probability of Occurring 0.0005 0.009 0.046  0.115 0.198 0.245 0.206 0.117 0.048 0.010  0.0006

Logic of Statistical Testing Inferring from samples – INFERENTIAL STATISTICS Scientists collect data from a sample and determine whether or not that sample provides EVIDENCE AGAINST the null hypothesis. If the null hypothesis is true, what is the probability we would have randomly chosen a sample with the values we observed? Analysis: By looking at our probability of obtaining 80% or 8 heads in a sample of 10 flips, we can make a decision. PROBABILITY IS OFTEN CALLED THE P value

Our Example Likelihood of getting 8 heads out of ten in our sample if the null hypothesis were actually true is p=0.0586 meaning it would occur roughly 6 times out of 100. Do you consider this value low or high? Do you think it provides enough evidence against the null hypothesis?

Statistical Significance Need a cut point for the p-value Common “cut points”: 0.05, 0.01, .001 If P value < 0.05, you say the result is “statistically significant” and you reject the null hypothesis (Ho). If the null hypothesis is true, the probability of randomly getting the observed sample is unlikely. This provides evidence against the null hypothesis and we would REJECT the null hypothesis, suggesting one of the alternative hypotheses were correct.

Statistical Significance If P value > 0.05, You say the results were “not statically significant” If the null hypothesis is true, the probability of randomly getting the observed sample is likely. This does not provides evidence against the null hypothesis and we would FAIL TO REJECT OR ACCEPT the null hypothesis, allowing us to reject the alternative hypotheses.

Statistical Tests/Hypothesis Testing/Inferential Test: All statistical tests provide a P value that is the probability that your results would have occurred if the null hypothesis were true. They use information from your data (mean, standard deviation, etc.) to figure out a probability based upon a population that meets the null hypothesis (much like our card simulation). You use the p-value to make a data-driven decision

Example Hypotheses and P value Null Hypothesis P-Value Decision Interpretation The mean life-span is 15 years. 0.078 The correlation between amount of nutrient and growth is 0. The mean height of plants exposed to sunlight equals the mean height of plants not exposed to light. Cut-off value is 0.05

Example Hypotheses and p-value Null Hypothesis P-Value Decision Interpretation The mean life-span is 15 years. 0.078 Do not reject the null hypothesis There is no evidence to suggest the mean life-span is not 15 years. The correlation between amount of nutrient and growth is 0. The mean height of plants exposed to sunlight equals the mean height of plants not exposed to light. Cut-off value is 0.05

Example Hypotheses and p-value Null Hypothesis P-Value Decision Interpretation The mean life-span is 15 years. 0.078 Do not reject the null hypothesis There is no evidence to suggest the mean life-span is not 15 years. The correlation between amount of nutrient and growth is 0. 0.010 The mean height of plants exposed to sunlight equals the mean height of plants not exposed to light. Cut-off value is 0.05

Example Hypotheses and p-value Null Hypothesis P-Value Decision Interpretation The mean life-span is 15 years. 0.078 Do not reject the null hypothesis There is no evidence to suggest the mean life-span is not 15 years. The correlation between amount of nutrient and growth is 0. 0.010 Reject the null hypothesis (P < 0.05) There is evidence to suggest the correlation is not zero. The mean height of plants exposed to sunlight equals the mean height of plants not exposed to light. Cut-off value is 0.05

Example Hypotheses and p-value Null Hypothesis P-Value Decision Interpretation The mean life-span is 15 years. 0.078 Do not reject the null hypothesis There is no evidence to suggest the mean life-span is not 15 years. The correlation between amount of nutrient and growth is 0. 0.010 Reject the null hypothesis (at sig level of 0.05) There is evidence to suggest the correlation is not zero. The mean height of plants exposed to sunlight equals the mean height of plants not exposed to light. 0.0001 Cut-off value is 0.05

Example Hypotheses and p-value Null Hypothesis P-Value Decision Interpretation The mean life-span is 15 years. 0.078 Do not reject the null hypothesis There is no evidence to suggest the mean life-span is not 15 years. The correlation between amount of nutrient and growth is 0. 0.010 Reject the null hypothesis (at sig level of 0.05) There is evidence to suggest the correlation is not zero. The mean height of plants exposed to sunlight equals the mean height of plants not exposed to light. 0.0001 Reject the null hypothesis There is evidence to suggest light makes a difference on plan growth. Cut-off value is 0.05