Question

he score distribution shown in the table is for all students who took a yearly AP statistics exam. An AP statistics teacher had 41 students preparing to take the AP exam. Though they were obviously not a random​ sample, he considered his students to be​ "typical" of all the national students.​ What's the probability that his students will achieve an average score of at least​ 3? Score Percent of students 5 12.1 4 22.6 3 24.3 2 18.1 1 22.9 The probabability that his students will achieve an average score of at least 3 is nothing . ​(Round to four decimal places as​ needed.).

Answer 1

X	P(X)	X*P(X)	X² * P(X)
5	0.121	0.605	3.025
4	0.226	0.904	3.616
3	0.243	0.729	2.1870
2	0.181	0.362	0.7240
1	0.229	0.2290	0.2290

mean = E[X] = Σx*P(X) =            2.82900
          
E [ X² ] = ΣX² * P(X) =            9.7810
          
variance = E[ X² ] - (E[ X ])² =            1.7778
          
std dev = √(variance) =            1.3333
==================

µ =    2.829                                      
σ =    1.333
n=   41                                      
                                          
X =   3                                      
                                          
Z =   (X - µ )/(σ/√n) = (   3   -   2.829   ) / (    1.333 / √   41   ) =   0.821  
                                          
P(X ≥   3   ) = P(Z ≥   0.82   ) =   P ( Z <   -0.821   ) =    0.2058           (answer)

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