Question

Suppose a fitness center has two weight-loss programs. Fifteen students complete Program A, and fifteen students complete Program B. Afterward, the mean and standard deviation of weight loss for each sample are computed (summarized below). What is the difference between the mean weight losses, among all students in the population? Answer with 95% confidence.

Prog A - Mean 10.5 St dev 5.6

Prog B - Mean 13.1 St dev 5.2

Answer 1

There are two weight loss programs , A and B.

Given that :

Program-----	A	B
Sample Size (n)	15	15
Mean ()	10.5	13.1
Std. Deviation (S )	5.6	5.2

We find if there is any difference between the means of two programs among students in the population ( _A -  _B ) using a 95% Confidence Interval.

So, we use a  two-sample t test

Test Statistic :  ( ( _A - _B ) - ( _A -  _B ) ) / (Sp ( 1/n_A   + 1/n_B )^0.5 ) ~ t _{nA + nB - 2}

where , S_p² is the pooled variance and is calculated as :

S_p²   = ( ( n_A-1 ) S_A² +  ( n_B -1 ) S_B² ) / (  n_A + n_A  - 2 )

Calculations  :   S_p² = (14 (5.6²) + 14 (5.2²) ) / 28 = 29.2

P ( - t_28,0.025 < ( ( 10.5-13.1 ) - ( _A -  _B ) ) / ( 29.2^0.5 * ( 1/15 + 1/15 )^0.5 ) < t_28,0.025 ) = 0.95

Since , t_28,0.025 = 2.048 ( Using t- distribution tables )

   P ( -2.048 < ( - 2.6 - ( _A -  _B ) ) / 1.97315 ) < 2.048 ) = 0.95

P ( -4.041 < ( - 2.6 - ( _A -  _B ) ) < 4.041 ) = 0.95

   P ( -1.441 <  - ( _A -  _B ) < 6.641 ) = 0.95

        ( _A -  _B )   ( - 6.641 , 1.441 )

Since , Zero is included in the above C.I , we have sufficient evidence that there is no significant difference between the means of the two weight loss programs among the students in the population.