Question

In: Statistics and Probability

1.       A   university   researcher   is   interested   in   whether   recent   recruitment   efforts   have   changed&

1.       A   university   researcher   is   interested   in   whether   recent   recruitment   efforts   have   changed   the   type   of   students   admitted   to   the   university.   To   test   this,   she   randomly   selects   50   freshmen   from   the   university   and   records   their   high   school   GPA.   The   mean   is   2.90   with   a   standard   deviation   of   0.70.   The   researcher   also   knows   that   the   mean   high   school   GPA   of   all   freshmen   enrolled   at   the   university   five   years   ago   was   2.75   with   a   standard   deviation   of   0.36.   The   researcher   wants   to   know   if   the   high   school   GPA   of   current   freshmen   at   the   university   is   different   than   that   of   freshmen   from   five   years   ago.  
   (a)   What   are   the   null   and   alternative   hypotheses   in   this   study   (stated   mathematically)?  
   (b)   Should   the   researcher   use   a   one-tailed   or   a   two-tailed   test?  
   (c)   Compute   the   appropriate   test   statistic   for   testing   the   hypothesis.  
   (d)   Using   α   =   0.05,   what   do   you   conclude   about   the   high   school   GPA   of   current   freshman?   Be   sure   to   include   a   discussion   of   the   critical   value   in   your   answer.  
   (e)   What   type   of   error   might   the   researcher   be   making   in   part   (d)?  
  
   2.   A   researcher   believes   that   smoking   worsens   a   person’s   sense   of   smell.   To   test   this,   he   takes   a   sample   of   25   smokers   and   gives   them   a   test   of   olfactory   sensitivity.   In   this   test,   higher   scores   indicate   greater   sensitivity.   For   his   sample,   the   mean   score   on   the   test   is   15.1   with   a   standard   deviation   of   1.2.   The   researcher   knows   the   mean   score   in   the   population   is   15.5,   but   the   population   standard   deviation   is   unknown.  
   (a)   What   are   the   null   and   alternative   hypotheses   in   this   study   (stated   mathematically)?  
   (b)   Should   the   researcher   use   a   one-tailed   or   a   two-tailed   test?  
   (c)   Compute   the   appropriate   test   statistic   for   testing   the   hypothesis.  
   (d)   Using   α   =   0.01,   do   you   conclude   that   smoking   affects   a   person’s   sense   of   smell?   Be   sure   to   include   a   discussion   of   the   critical   value   in   your   answer.  
   (e)   What   type   of   error   might   the   researcher   be   making   in   part   (d)?  
  

Solutions

Expert Solution

1)

Ho :   µ =   2.75
Ha :   µ ╪   2.75

b) two tailed

c)

Level of Significance ,    α =    0.050                  
population std dev ,    σ =    0.3600                  
Sample Size ,   n =    50                  
Sample Mean,    x̅ =   2.9000                  
                          
'   '   '                  
                          
Standard Error , SE = σ/√n =   0.3600   / √    50   =   0.0509      
Z-test statistic= (x̅ - µ )/SE = (   2.900   -   2.75   ) /    0.0509   =   2.946

d)

critical z value, z* =   ±   1.960
      
Decision: test stat > 196  Reject null hypothesis   
Conclusion: There is enough evidence to conclude that  high   school   GPA   of   current   freshmen   at   the   university   is   different   than   that   of   freshmen   from   five   years   ago.

e)

since, we reject the null hypothesis, so, there might be type I error


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