9.1 – The Basics Ch 9 – Testing a Claim

Jack’s a candidate for mayor against 1 other person, so he must gain at least 50% of the votes. Based on a poll of voters just before the election, can Jack be confident of victory? We will do this again with a different bag with a different proportion of white chips. 1 Random Selection of a chip. White chip = Voting for Jack 2 Construct Scatter plot: X-axis = # of chips Y-axis = proportion of white 3 Keep drawing until your confident Jack will win

Two important Aspect of Hypothesis Testing: 1. increasing n help us to be more confident that there is a difference between what is observed and what is claimed. 2. It is easier to reject what is claimed if the true parameter is farther away from what is hypothesized

Goal of the Two Inferences Confidence Intervals - Estimate a population parameter Significance Test - Assess the evidence provided by data about some claim concerning a population

Suppose a random sample of 100 voters shows that 56 will vote for Jack. Is his confidence warranted? Maybe so, but there are two explanations of why the majority of voters in the sample seem to favor Jack. Should Jack be confident in a victory? Reasons: 1.In reality less than 50% favor Jack, and the sample result was just do to sampling variability. If this is true, then Jack shouldn’t be confident of a victory. 2.The sample is above 50% because more than 50% will actually vote for Jack. Then he should believe this sample and be confident of victory ONLY IF he can rule out the first reason!

A Fathom Simulation was done to simulate 400 samples of size 100 from the population in which exactly 50% of the voters support Jack. 55 of the 400 trials had a sample proportion of 56% or higher. Jack should be worried of winning! 13.75% is a high chance that these results would have happened by pure chance alone if the true value is 50%. It is plausible that 50% of the voters do not favor Jack.

Null Hypothesis – H 0 : this is the claim we are testing. This is typically a statement of no difference. H 0 : (parameter symbol) = ## Alternate Hypothesis – Ha : this is the claim about the population that we are trying to find evidence for This statement is one-sided or two-sided. H a : (parameter symbol) > ## H a : (parameter symbol) < ## H a : (parameter symbol) = ## Stating our Hypotheses

ALWAYS STATE THE HYPOTHESIS BEFORE USING THE DATA!!!! HYPOTHESES ARE ABOUT A POPULATION. YOUR EVIDENCE IS YOUR SAMPLE.

» Assuming the Ho is true, the probability the p-hat of x- bar would take on a value more extreme than the 1 observed » Smaller = the more rare our sample results are and the more evidence we have to support Ha and against Ho » Larger = the more likely our sample results our and the less evidence we have to support Ha and we fail to reject Ho. P-VALUE

» If exactly 50% support Jack, there is a 13.75% chance in a sample of 100, 56% or more would actually support him by random chance alone. » Since this is not a rare occurrence, Jack should be worried that 50% actually support him. Interpret the P-value for Jack’s chances of winning

Mike tries a new 7-iron. He has years of experience and knows his mean is 175 yards and his standard deviation is 15 yards using his old club. He hope this new club will make him more consistent. a)Describe the parameter of interest. b)State the hypotheses for performing a significance test.

From 50 shots, his sample standard deviation is 10.9 yards. A significance test was performed and the p-value is a)Interpret the P-value. b)Does the data provide convincing evidence against Ho? Explain

» Statistically Significant level – alpha » Stated BEFORE the test is conducted. » Typically 0.05 (or 0.01) it depends on the context and severity of the results. How low is low?