Question

A biologist observes a sample of 25 cells to have a mean dividing time of 30 minutes with a standard deviation of 3.2 minutes.

a. Construct a 99% confidence interval for population mean dividing time.

b. A colleague claims that the mean dividing time is 32 minutes, but the biologist believes that it is less than 32 minutes. Carry out a hypothesis test and decide who is right - the biologist or the colleague.

c. State your conclusion in context of the hypothesis test and confidence interval.

d. What is the possible error made? What does the error mean in this problem situation?

Answer 1

a)

Level of Significance ,    α =    0.01          
degree of freedom=   DF=n-1=   24          
't value='   tα/2=   2.797   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   3.2/√25=   0.6400          
margin of error , E=t*SE =   2.7969   *   0.6400   =   1.7900
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    30.00   -   1.790041   =   28.2100
Interval Upper Limit = x̅ + E =    30.00   -   1.790041   =   31.7900
99%   confidence interval is (   28.21   < µ <   31.79   )

b)

Ho :   µ =   32  
Ha :   µ <   32   (Left tail test)
          
Level of Significance ,    α =    0.010  
sample std dev ,    s =    3.2000  
Sample Size ,   n =    25  
Sample Mean,    x̅ =   30.0000  
          
degree of freedom=   DF=n-1=   24  
          
Standard Error , SE = s/√n =   3.2/√25=   0.6400  
t-test statistic= (x̅ - µ )/SE =    (30-32)/0.64=   -3.125  
          
critical t value, t* =        -2.492   [Excel formula =t.inv(α/no. of tails,df) ]
          
p-Value   =   0.0023   [Excel formula =t.dist(t-stat,df) ]
Decision:   p-value<α, Reject null hypothesis       
Conclusion: There is enough evidence that biologist is correct

c)

  There is enough evidence to reject the null hypothesis that true mean is 32 minutes

there is 99% confidence that true ,mean lies within confidence interval

d)

Type I error could be made

Type I error is concluding that mean is less than 32 but infact it is 32 minutes