Hypothesis Testing Introduction

What is a Hypothesis Test A hypothesis test is a statistical test that determines if there’s a STATISTICALLY SIGNIFICANT difference between: ◦ A claimed population mean & sample mean ◦ An ideal average & an observed average ◦ An entire distribution ◦ Or a significant relationship between 2 variables.

So What’s Statistically Significant? Say the average tiger shark is supposed to weigh 1250 lbs. When weigh a random sample of tiger sharks, their average isn’t likely to be 1250 lbs. A hypothesis test will help you decide if you’re sample is too wacky for the average of 1250 to be correct. Statistically Significant means the SAMPLE data you obtained is SOO ABNORMAL it couldn’t have just happened by random chance. Think about it…

So How Do We Do It? Following are the steps for each Hypothesis test: 1.Develop & write hypotheses 2.Decide on correct test and check the conditions 3.Find the correct critical value and obtain a p-value 4.Make a contextual decision about your hypotheses

Step 1: Writing Hypotheses These depend on the type of data you’re working with, but there are a few generic rules: 1.There are 2 Hypotheses: 1.The null (H 0 ) 2.The alternate (H a )

Quick Glance at the Hypotheses Null Hypothesis (H o ) ◦S◦S ays there is NO effect or NO change in the population ◦T◦T he null would be the “ideal” average Alternate Hypothesis (H a ) ◦T◦T he Alternative to no effect or no change ◦O◦O ften stated as “greater than” or “less than” the population A hypothesis test sets out to decide between these 2 hypotheses

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 Billy Blanks claims anyone that uses his program will lose 10 lbs in 3 weeks, with a standard deviation of 2.35lbs. As the founder of BaeToe, you are out to prove that Billy’s customers lose less poundage than his claim. If you can prove that, you feel like more people will buy your DVD’s!! H o :  = 10 lbs.(The pop mean lbs lost to a BB DVD is 10 lbs.)

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 way you are trying to disprove the H 0 Billy Blanks claims anyone that uses his program will lose 10 lbs in 3 weeks, with a standard deviation of 2.35lbs. As the founder of BaeToe, you are out to prove that Billy’s customers lose less poundage than his claim. If you can prove that, you feel like more people will buy your DVD’s!! H a :  < 2.35 lbs (The pop mean of pounds lost to Billy Blanks’ DVDs is less than 10 lbs.)

Flow Rate The Association for Fish Growers Anonymous claims the ideal flow rate in a large tank for growing fish is 22.5 Liters/min. To test his tank’s effectiveness, Frankie takes an SRS of 50 flow rates from the last two days. He calculates an average flow rate of 19.2 Liters/min with a standard deviation of 1.3 L/m. Is Frankie’s flow rate significantly less than the ideal rate? H o : Jake’s average flow rate is the ideal flow rate of 22.5 L/m. μ = 22.5 H a : Jake’s average flow rate is significantly less than the ideal flow rate of 22.5 L/m. μ < 22.5

Step 2: Identify the Type of Test & Check the Conditions We’ll look at 3.5 types of tests for research (there are more!!): ◦ T-Testing  2 – Sample T Testing ◦ Chi Square Testing ◦ Linear Regression T-Testing Flow Rates, Fish Masses, Other Avgs… T-Testing 2-Sample T-Testing Water Quality Tests, Comparing 2 Water Tanks, Comparing types of fish, treatments, seasons, etc… Chi Square Testing Genetics, Punnett Squares Lin Reg T-Testing Bay grass growth investigations, Treatment tests, etc…

Step 3: Finding the Critical # and the p-value The critical # leads to the decision value and varies depending on the type of test. The critical values are: ◦ T for t-tests ◦ Χ 2 for Chi Square Tests ◦ T for Linear Regression tests

P-Value This value is the basis for our conclusions about the Null Hypothesis (found on charts) This value is the probability to the right of the observed x-bar value for > and left for < The p-value is the probability of getting our result (sample) if the hypothesized mean is true. 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

2 Types of Tests 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 (≠)

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

Step 4: Making a Decision Compare your p-value to a significance level ◦ Significance Level – pre-determined level of accuracy that serves as the cutoff point for acceptable values ( α – alpha level)  Typically α =.05, but another value may be given/used

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) Want to be Statistically Significant? Have a p-value LESS than 0.05

Significance Level & Decisions Significance Level (  ) ◦ Predetermined value that becomes the decisive value that determines your rejection of the H o (If not otherwise specified  *P-value >  Fail to Reject the Null Hypothesis **P-value ≤  Reject the Null Hypothesis; Accept the Alternate Statistical Significance P-value ≤ 

What’s a Test Look Like? 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 p =.0016 What does this mean? z = There’s only a.16% chance that our sample would happen if their population mean is correct. Since p is less than alpha, we reject the Null Hypothesis and accept that the average battery lasts less than 22.3 hours.

P-Decision Chart P-value vs. α-level Decision p < α Reject H o : Accept H a p > α Fail to Reject H o Since we never know the TRUE ideal value or population value we NEVER accept the H a. We only say we don’t have statistically significant evidence to reject it.

Adding Context Make sure you relate your decision to the context of the problem/experiment Making a contextual decision is 80% of the battle… Ex: Since our p-value is less than our significance level, we have statistically significant evidence to reject the water quality is the same between the two fish tanks and can accept that the treatment tank has a higher water quality.

Hypothesis Test Steps Review 1.Develop & write hypotheses 2.Decide on correct test and check the conditions 3.Find the correct critical value and obtain a p-value 4.Make a contextual decision about your hypotheses