Question

4. The following are 20 observations of response times (in seconds) from a random sample of participants on a cognitive psychology task:

1.7       0.8       4.3       2.9       2.3       1.1       2.2       1.8       2.0       1.2       4.4       1.6       3.8       1.5       2.8

3.3       1.8       2.5       2.7       1.6

(a) Calculate the mean and unbiased standard deviation of the response times.

(b) Construct and interpret a 95% confidence interval for μ, which is the true mean response time. Assume that σ = 1.5.

(c) Construct and interpret a 90% confidence interval for μ, which is the true mean response time. Assume that σ is unknown.

(d) What would happen to the width of your confidence interval if our sample size increased? Explain.

Answer 1

a)

Sample Mean,    x̅ = ΣX/n =    2.3150
sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   1.0246

b)

Level of Significance ,    α =    0.05          
'   '   '          
z value=   z α/2=   1.960   [Excel formula =NORMSINV(α/2) ]      
                  
Standard Error , SE = σ/√n =   1.5000   / √   20   =   0.3354
margin of error, E=Z*SE =   1.9600   *   0.3354   =   0.6574
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    2.32   -   0.657392   =   1.6576
Interval Upper Limit = x̅ + E =    2.32   -   0.657392   =   2.9724
95%   confidence interval is (   1.6576   < µ <   2.9724   )

c)

Level of Significance ,    α =    0.1          
degree of freedom=   DF=n-1=   19          
't value='   tα/2=   1.729   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   1.0246   / √   20   =   0.2291
margin of error , E=t*SE =   1.7291   *   0.2291   =   0.3961
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    2.32   -   0.396149   =   1.9189
Interval Upper Limit = x̅ + E =    2.32   -   0.396149   =   2.7111
90%   confidence interval is (   1.9189   < µ <   2.7111   )

d)

width of your confidence interval will get decreased if our sample size increased