Question

40 individuals were asked the following question: What is your favorite soft drink? The table below shows the results.

Soft Drink	How many times a week do you typically exercise?
Mountain Dew	7
Coke	3
Diet Coke	2
Dr. Pepper	10
Sprite	8
Diet Mountain Dew	1
Lemonade	3
Diet Pepsi	2
Root Beer	3
Pepsi	1

1. Calculate the sample proportion for individuals that answered “Dr. Pepper”. SHOW THE CALCULATION

2. Calculate the appropriate SE based on the 40 individuals and #1 above. SHOW THE CALCULATION.

3. Calculate a 95% confidence interval for the proportion associated with the individuals who answred “Diet Coke”. SHOW ALL CALCULATIONS.

4. Interpret the full meaning of the confidence interval you have in #12. Remember to include all four components to an interpretation

Answer 1

1)

sample proportion for individuals that answered “Dr. Pepper” = 10/40 = 0.25

2) Standard Error ,    SE = √[p̂(1-p̂)/n] = √(0.25*0.75/40) = 0.06847

3)

Level of Significance,   α =    0.05          
Number of Items of Interest,   x =   2          
Sample Size,   n =    40          
                  
Sample Proportion ,    p̂ = x/n =    0.0500          
z -value =   Zα/2 =    1.960   [excel formula =NORMSINV(α/2)]      
                  
Standard Error ,    SE = √[p̂(1-p̂)/n] =    0.03446          
margin of error , E = Z*SE =    1.960   *   0.03446   =   0.0675
                  
95%   Confidence Interval is              
Interval Lower Limit = p̂ - E =    0.05000   -   0.06754   =   -0.01754
Interval Upper Limit = p̂ + E =   0.05000   +   0.06754   =   0.11754
                  
95%   confidence interval is (   -0.018   < p <    0.118   )

4)

There is 95% confidence that true proportion of Diet Coke lies within confidence interval