Question

The proportion is 14/40 students play sports.

A.- Estimate with 95% confidence the percent of students playing a sport at School(rec., club, or official team). Check the necessary conditions and interpret your confidence interval. No matter how you obtained your sample, pretend that it is a random sample for now.

B.

Use the data obtained in part 1 to answer the question: Is the percent of students playing a sport different than 50%? Complete all the usual steps.

Answer 1

a)

since, np = 40 * 14/40 = 14 ≥5

and n(1-p) = 36 ≥5

also, assuming n=40 <0.05N

so, all necessary conditions are met

Level of Significance,   α =    0.05
Number of Items of Interest,   x =   14
Sample Size,   n =    40
      
Sample Proportion ,    p̂ = x/n =    0.350

z -value =   "Zα/2 =
"   1.9600   [excel formula =NORMSINV(α/2)]
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.0754  
          
margin of error ,    E = Z*SE =    0.1478  
          
Confidence Interval          
Interval Lower Limit , =    p̂ - E =    0.202  
Interval Upper Limit , =    p̂ + E =   0.498  

so, confidence interval is ( 20.2% < p < 49.8% )

there is 95% confidence that true percent of students playing a sport at School(rec., club, or official team) lies within confidence interval

b)

Ho :   p = 50%
H1 :   p ╪ 50%

since, confidence interval do not contain null hypothesis 50% , so, reject null hypothesis,

so, there is enough evidence to conclude that true percent of students playing a sport different than 50% with 95% confidence