Tests of Significance: Stating Hypothesis; Testing Population Mean

Tests of Significance Use a confidence interval to… estimate a population parameter (µ) Use a Test of Significance to… Assess the evidence provided by data about some claim concerning a population Use a confidence interval to… estimate a population parameter (µ) Use a Test of Significance to… Assess the evidence provided by data about some claim concerning a population Tim claims he can run a mile in under 6 minutes 80% of the time. You make Tim run a mile while you watch and it takes him 20 minutes. You then call him a liar!!! You would use a significance test to determine the probability that he would run a 20 minute mile if his claim of 6 minutes was actually true which would either prove or disprove his statement.

Breaking Down a Significance Test The question a significance test asks is: Does the sample result reflect something that is a true result of the experiment OR Is the result simply a result of CHANCE? The question a significance test asks is: Does the sample result reflect something that is a true result of the experiment OR Is the result simply a result of CHANCE? We will investigate this question by investigating two Hypotheses: The Null Hypothesis (H o )The Alternative Hypothesis (H a )

Quick Glance at the Hypotheses Null Hypothesis (H o ) Says there is NO effect or NO change in the population Alternative Hypothesis (H a ) The Alternative to no effect or no change Often stated as “greater than” or “less than” the population Null Hypothesis (H o ) Says there is NO effect or NO change in the population Alternative Hypothesis (H a ) The Alternative to no effect or no change Often stated as “greater than” or “less than” the population We are set out to decide between one of these Hypotheses…

P-Value This value is the basis for our conclusions about the Null Hypothesis This value is the probability of being more extreme than the observed x-bar value This value is the basis for our conclusions about the Null Hypothesis This value is the probability of being more extreme than the observed x-bar value The p-value is the probability of a result at least as far out as the result we got. The smaller the p-value, the stronger the evidence AGAINST the H o. Why? The less likely your sample is to occur given H o

More About P Small p-values : p<.05 Give evidence against H o because they say the observed result is unlikely to occur by chance (accept H a ) Large p-values : p>.05 Fail to give evidence (or reject) against the H o (we never actually accept H o, we just fail to reject it) Small p-values : p<.05 Give evidence against H o because they say the observed result is unlikely to occur by chance (accept H a ) Large p-values : p>.05 Fail to give evidence (or reject) against the H o (we never actually accept H o, we just fail to reject it) Want to be Statistically Significant? Have a p-value LESS than 0.05

The Outline of a Significance Test 1. Describe the desired effect in terms of a parameter (i.e. -  ) 2. Write the Hypotheses 3. Calculate a statistic (x-bar) that estimates the parameter 4. Find the p-value for that statistic and decide if you should reject or fail to reject the null 1. Describe the desired effect in terms of a parameter (i.e. -  ) 2. Write the Hypotheses 3. Calculate a statistic (x-bar) that estimates the parameter 4. Find the p-value for that statistic and decide if you should reject or fail to reject the null

Stating Hypotheses The Null Hypothesis (H o ) The statement being tested in a test of significance Statement of “no effect” or “no difference” We are testing the strength of the evidence AGAINST the H o The Null Hypothesis (H o ) The statement being tested in a test of significance Statement of “no effect” or “no difference” We are testing the strength of the evidence AGAINST the H o Census Bureau data show that the mean household income in the area served by Turfland Mall is $42,500 per year. A market research firm suspect the mean household income of mall shoppers is higher that of the general population. H o :  = $42,500 (The population mean is $42,500 per year)

Stating Hypotheses The Alternative Hypothesis (H a ) The part of the claim you are investigating Statement varies depending on problem Either 1-sided (≥,≤, ) or 2-sided (≠) The Alternative Hypothesis (H a ) The part of the claim you are investigating Statement varies depending on problem Either 1-sided (≥,≤, ) or 2-sided (≠) Census Bureau data show that the mean household income in the area served by Turfland Mall is $42,500 per year. A market research firm suspect the mean household income of mall shoppers is higher that of the general population. H a :  > $42,500 (The population mean greater than $42,500 per year)

Duracell Batteries Duracell claims the average lifespan of their Ultra Coppertop C Batteries is 22.3 hours with a standard deviation of 23 minutes. You take a SRS of 200 batteries and find the mean to be 21.4 hours. At a 5% significance level, is there enough evidence to refute Duracell’s claim? H o : The mean lifetime of a Duracell Ultra Coppertop C Battery is 22.3 hours. µ = 22.3hrs H a : The mean lifetime of a Duracell Ultra Coppertop C Battery is less than 22.3 hours. µ < 22.3hrs

Deciding Between 1 or 2 Sided Testing One Sided Examination (stated in problem) is looking for greater than, less than, or some specific side of the H o Two Sided Examination (stated in problem) is vague; just looking at not equal to the mean, but not mentioning a specific side of the H o One Sided Examination (stated in problem) is looking for greater than, less than, or some specific side of the H o Two Sided Examination (stated in problem) is vague; just looking at not equal to the mean, but not mentioning a specific side of the H o

Investigating the P-Value P value is the probability that a value as extreme or worse than what was observed could actually happen Smaller the P-value, the stronger evidence AGAINST the H o P value is the probability that a value as extreme or worse than what was observed could actually happen Smaller the P-value, the stronger evidence AGAINST the H o Find the Sample Mean, find the z-score, and then find the appropriate area to represent p. ≠ < or ≤> or ≥

Extreme and Significant Significance Level (  ) Predetermined p-value that becomes the decisive value that determines your rejection of the H o Often  =.05 Significance Level (  ) Predetermined p-value that becomes the decisive value that determines your rejection of the H o Often  =.05 Statistical Significance P-value ≤ 

Inference for population mean Identify the population of interest & the parameter you want to draw conclusions about. State the null and alternative hypothesis Choose the appropriate inference procedure and verify the conditions. If the conditions are met, carry out the procedure Calculate the test statistic (z-value) Find the p-value and compare to  Interpret your results in context Identify the population of interest & the parameter you want to draw conclusions about. State the null and alternative hypothesis Choose the appropriate inference procedure and verify the conditions. If the conditions are met, carry out the procedure Calculate the test statistic (z-value) Find the p-value and compare to  Interpret your results in context

Inference for population mean (z-test for population mean Conditions needed for this procedure: Testing H 0 : µ = µ 0 Known σ and SRS from sample of size n Now, let’s look at the components of the procedure: Conditions needed for this procedure: Testing H 0 : µ = µ 0 Known σ and SRS from sample of size n Now, let’s look at the components of the procedure: Test Statistic (z-score) Let’s look at how to find the p- value…

Finding the right p The p-value is based on the H a H a : µ > µ 0 H a : µ < µ 0 H a : µ ≠ µ 0 One-sample tests 2-sided test

Duracell Batteries Duracell claims the average lifespan of their Ultra Coppertop C Batteries is 22.3 hours with a standard deviation of 23 minutes. You take a SRS of 200 batteries and find the mean to be 21.4 hours. At a 5% significance level, is there enough evidence to refute Duracell’s claim? Ho: µ = 22.3hrs Ha: µ ≤ 22.3hrs Does their mean fit in our CI?

Z Tests for µ with Fixed Significance Level (alpha) With these tests you are given an alpha level against which you test your p- value p ≤  – Reject the null; accept the H a p >  – Fail to reject the null With these tests you are given an alpha level against which you test your p- value p ≤  – Reject the null; accept the H a p >  – Fail to reject the null Instead of comparing to the standard  =. 05, we use the significance level the problem requires

Confidence Interval & 2-Sided Tests A two sided test is just looking to see if your value fits into a parallel confidence interval For a 2 sided z-test… A two sided test is just looking to see if your value fits into a parallel confidence interval For a 2 sided z-test…  =.05 for 2 sided has.025 on each side But… So does a 95% Confidence Interval So for a 2 sided z test... You could construct a CI to test for significance And test to see if µ 0 fits in your CI… If not, reject H 0

Duracell Batteries Duracell claims the average lifespan of their Ultra Coppertop C Batteries is 22.3 hours with a standard deviation of 23 minutes. You take a SRS of 200 batteries and find the mean to be hours. At a 5% significance level, is there enough evidence to show the average is less than Duracell’s claim? Ho: µ = 22.3hrs Ha: µ ≤ 22.3hrs z = p = What does this mean?

Homework #43,44,46-49,54,55 Next class – testing on calculator and decisions #43,44,46-49,54,55 Next class – testing on calculator and decisions