II. Two Approaches to Testing Hypotheses about Differences between and "a"

• Confidence Intervals
  
  • Does CI for include the specific value "a" (the value is being compared to)?
    
    
    • If yes,

      
      
      
      
      
      

    • If no,

      
      
      
      
      
      

• Formal Hypothesis Testing (z- or t-test)
  
  
  • Is sample mean significantly different from a specific value "a"?
    
    
  • Distribution choice:
    • use z-distribution if sample size is _________ (no df)
      
      
    • use t-distribution if sample size is __________(df changes with _____)

III. **Steps in Hypothesis Testing**

Step 1: State Hypotheses (2 types)

• Hypotheses

1) Alternative -

• m for the pop from which sample was drawn is‚ , < or > the value "a"
  

• H_A:

• "a" is a constant value derived from knowledge about the situation.
  
  
• Will be accepted (tentatively) only if null is very unlikely to be true.
  

2) Null -

• for the pop from which the sample was drawn is equal to the value "a"
  

• H₀:
  
  
  
• Will be accepted (tentatively) until there is evidence that it is very unlikely to be true.
  
  
  
  

 

• Alternative and Null must together cover all possible outcomes of results.

If					Then
	HA:	m a				H0:	m
	HA:	m < a				H0:	m
	HA:	m > a				H0:	m

 

• Directionality of Hypotheses
  
  
  • Nondirectional
    
    • if can't predict whether m > or < "a"

      
      
      
      
      
      

  • Directional
    
    • if can predict whether m > or < "a"
      
      • H_A:

        
        
        

        
        

      • H_A:

        
        

• Decisions are based on Null
  
  • We will choose (accept) H₀ if could have deviated the observed amount from "a" by chance.

    
    

  • We will not choose (reject) H₀ and will tentatively choose (accept) H_A if it is unlikely that deviated the observed amount from "a" just by chance alone.

    
    
    
    
    
    

Step 2: Make a graph

• Draw a Sampling Distribution for , assuming = a
  

Step 3: Find the Critical Value(s) (CV)

• CV is the value with a (if directional H_A) or a /2 (if nondirectional H_A) beyond it.
  
• found in z- or t-table
  
• called z_crit or t_crit
  
• CV located at the boundary of the rejection region (RR)
  
  
• label CV(s), & rejection region on graph!
  

  
  
  
  
  
  
  
  
  
  
  
  
  

Step 4: State the Decision Rules formally

• we will reject the null if

  
  
  
  

• we will not reject the null (i.e. will accept it) if

Step 5: Calculate the Test Statistic

• z- or t-value is calculated from the sample data
  
  
• called z_calc or t_calc
  
  
• general form of z & t Test Statistics:
  

Sample Statistic - "a"
Standard Error of Statistic

• computational formulas:

Step 6: State Assumptions of the test

• Subjects were randomly and independently selected from the population
  
  
• Sampling Distribution of the mean is normal in form
  
  
  • it is normal if either of the following are true
    
    
    • 
      
      
      
      
      
    • 
      
      
      
      
      
      
• If these assumptions are violated, then test results will not be accurate.
  
  

Step 7: Draw Conclusion

• Reject H₀ (& tentatively accept H_A) if test statistic falls in rejection region.
  
  • unlikely to have deviated from "a" this much by chance alone.
    
    
    
    
  •  

• Do not reject H₀ if test statistic falls outside rejection region.
  
  • could have deviated from "a" this much by chance alone.

    
    

  •  

Sample Problems

1. A third grade teacher develops a new reading program. She expects her 75 students to score higher on an achievement test than the population of all third graders (for whom = 100). The mean of her students' achievement scores is 107 (& s = 15). Is her reading program effective?

• Hypotheses: alpha = .01




• Graph:












• Critical Value

 

•Decision Rule





• Test Statistic







•Assumptions






•Decision





•P-value

 

 

 

2. A researcher thinks that depth perception is affected by fatigue. A widely used task in which people are asked to estimate a standard distance between themselves and a standard object in a semi-darkened room is known to produce normally distributed responses, with a mean of 22 feet. Sixteen participants who perform the distance estimation task at the end of a day after participating in 2 hours of some taxing activity showed a mean of 25 feet (s = 16).

• Hypotheses: alpha = .01




• Graph:












• Critical Value

 

•Decision Rule





• Test Statistic







•Assumptions





•Decision





•P-value

 

 

 

 

3. A researcher thinks that participants are likely to use a different number of adjectives when describing themselves than they do when describing others. Previous research has shown that when asked to describe a close friend, the number of adjectives used is normally distributed with a mean of 9.0. Nine participants asked to describe themselves showed a mean of 6.0 (s = 2.0).

• Hypotheses: alpha = .05




• Graph:

 

 

• Critical Value

 

 

 

•Decision Rule





• Test Statistic







•Assumptions

 

•Decision

•P-value

 

 

4. A physician thinks she has developed an effective treatment for AIDS. She knows that the average length of life is 100 months after disease onset. The mean life span of her sample of 150 treated patients is 110 (s = 40). Did the treatment increase life expectancy?

• Hypotheses: alpha = .05




• Graph:

• Critical Value

 

 

 

 

•Decision Rule





• Test Statistic







•Assumptions

 

•Decision





•P-value

 

 

 

 

5. A physician thinks that taking vitamins every day will reduce the number of common colds people experience during a year. He knows that the population mean number of colds per year is 3.2. He randomly selects 200 people and has them take vitamins for a year and keep track of how many colds they get. The mean number of colds in the sample 2.9 (s = .8). Does vitamin consumption decrease cold frequency?

• Hypotheses: alpha = .10




• Graph:

 

• Critical Value

 

•Decision Rule





• Test Statistic







•Assumptions

 

•Decision





•P-value

 

 

 

 

 

6. A researchers wonders whether children who exhibit thumb-sucking behavior at age 6 feel less secure than 6 year-olds in general. Scores on a standardized test of interpersonal security for 6-year-olds are normally distributed, witha mean of 82. Four 6-year-olds who are showing thumb-sucking behavior showed a mean interpersonal security score of 74 (s = 24).

• Hypotheses: alpha = .10




• Graph:

 

• Critical Value

 

 

•Decision Rule





• Test Statistic







•Assumptions

 

•Decision





•P-value