Assignment #11 Single Sample t-Test 

What factor determines whether you should use a z-test or a t-test statistic for a hypothesis test? 

A sample of n = 16 individuals is selected from a population mean of : = 74. A treatment is administered to the individuals in the sample and, after treatment, the sample variance is found to be s2 = 64. 

1. If the treatment has a 3-point effect and produces a sample mean of M =
   77, is this sufficient to conclude that there is a significant treatment effect
   using a two-tailed test with ” = .05?
    
2. If the treatment has a 6-point effect and produces a sample mean of M =
   80, is this sufficient to conclude that there is a significant treatment effect using a two-tailed test with ” = .05?
    

A major corporation in the Northeast noted that last year its employees averaged : = 5.8 absences during the winter season (December to February). In an attempt to reduce absences, the company offered free flu shots to all employees this year. For a sample of n = 100 people who took the shots, the average number of absences this winter was M = 3.6 with SS = 396. Do these data indicate a significant decrease in the number of absences? Use a one-tailed test with ” = .05. 

On a standardized spatial skills task, normative data reveal that people typically get : = 15 correct solutions. A psychologist tests n = 7 individuals who have brain injuries in the right cerebral hemisphere. For the following data, determine whether or not right-hemisphere damage results in significantly reduced performance on the spatial skills task. Test with alpha (“) set at .05 with one tail. The data are as follows: 12, 16, 9, 8, 10, 17, 10. 

11.3 

11.4 

I. Assumptions for t-test A. Populations 

Single Sample t-test 

1. the population from which the sample is selected is normal 

1. One random sample (with replacement)
    
2. Data values
    
   1. Sample values known (mean, standard deviation)
       
   2. Population values (mean, standard deviation) not known
       
3. Diagramming your research (shows the whole logic and process of hypothesis testing)
    
   1. Draw a picture of your research design (see diagramming your research handout).
       
   2. There are always two explanations (i.e. hypotheses) of your research results, the
      wording of which depends on whether the research question is directional (one-tailed)
      or non-directional (two-tailed). State them as logical opposites.
       
   3. For statistical testing, ignore the alternative hypothesis and focus on the null hypothesis,
      since the null hypothesis claims that the research results happened by chance through
      sampling error.
       
   4. Assuming that the null is true (i.e. that the research results occurred by chance through
      sampling error) allows one to do a probability calculation (i.e. all statistical tests are nothing more than calculating the probability of getting your research results by chance through sampling error).
       
   5. Observe that there are two outcomes which may occur from the results of the probability calculation (high or low probability of getting your research results by chance, depending on the alpha (α) level).
       
   6. Each outcome will lead to a decision about the null hypothesis, whether the null is probably true (i.e. we then accept the null to be true) or probably not true (i.e. we then reject the null as false).
       
4. Hypotheses (i.e. the two explanations of your research results)
    
   1. Two-tailed (non-directional research question)
       
      1. Alternative hypothesis (H1): The independent variable (i.e. the treatment) does make a difference in performance.
          
      2. Null hypothesis (H0): The independent variable (i.e. the treatment) does not make a difference in performance.
          
   2. One-tailed (directional research question)
       
      1. Alternative hypothesis (H1): The treatment has an increased (right tail) or a
         decreased (left tail) effect on performance.
          
      2. Null hypothesis (H0): The treatment has an opposite effect than expected or no
         change in performance.
          
5. Determine critical regions (i.e. the z score boundary between the high or low probability of getting your research results by chance) using table A-23
    

A. Significance level (should be given or decided prior to the research; also called the 

1 

confidence, alpha, or p level) 

1. α or p = .05, .01, or .001 

1. One- or two-tailed test
    
   1. One-tailed: use the first row across the top
       
   2. Two-tailed: use the second row across the top
       
2. Degrees of freedom
    

1. df = n – 1
D. With degrees of freedom & one- or two-tailed α value, find the critical t value 

1. If two-tailed, then critical t value is ± t value
    
2. If one-tailed, then determine if critical t value is +t (right tail; expecting an
   increase) or -t (left tail; expecting a decrease)
    
3. Calculate t-test statistic
    
   1. General statistical test formula
      t = observed sample mean – hypothesized populational mean standard error
       
   2. Calculations
       
      1. Compute variance
         ∑????2 − (∑????)2 ????????
         s2= ???? or
         ????−1 ????????
          
      2. Compute standard error (average distance between sample & pop means)
         Note: (standard error is simply an estimate of the average sampling error which may occur by chance, since a sample can never give a totally accurate picture of a population
         √????2 ????
          
      3. Compute t-test statistic (i.e. calculates the probability of getting your research results by chance through sampling error)
         ????−μ ????????
          
   3. Compare the calculated t-score to the critical t-score & make a decision about the null hypothesis
       
      1. Reject the null (as false) and accept the alternative or
          
      2. Accept null (as true)
          
4. Reporting the results of a Single Sample t test
   “The treatment had a significant effect on (M = 25, SD = 4.22); t(18) = +3.00, p < .05, two-tailed.”
   2