Question

3. Take the mean and standard deviation of data set A calculated in problem 1 and assume that they are population parameters (μ and σ) known for the variable fish length in a population of rainbow trouts in the Coldwater River. Imagine that data set B is a sample obtained from a different population in Red River (Chapter 6 problem!). a) Conduct a hypothesis test to see if the mean fish length in the Red River population is different from the population in Coldwater River. b) Conduct a hypothesis test to see if the variance in fish length is different in the Red River population compared to the variance in the Coldwater population.

• Data set A: total= 677.98, mean= 67.798, n= 10, variance= 0.663084, std devition= 0.814299972

• Data set B: total= 574.24, mean=71.78, n=8, variance= 0.727143, std devition= 0.852726719

 Do not use excel function for p value.  Show all your work

Answer 1

Data set A: total= 677.98, mean= 67.798, n= 10, variance= 0.663084, std devition= 0.814299972

Data set B: total= 574.24, mean=71.78, n=8, variance= 0.727143, std devition= 0.852726719

a) Conduct a hypothesis test to see if the mean fish length in the Red River population is different from the population in Coldwater River.

Answer: here we have two independant group, so we will use the independant t-test

Independent t-test formula

• Let A and B represent the two groups to compare.
• Let mAmA and mBmB represent the means of groups A and B, respectively.
• Let nAnA and nBnB represent the sizes of group A and B, respectively.

The t test statistic value to test whether the means are different can be calculated as follow :

Once t-test statistic value is determined, you have to read in t-test table the critical value of Student’s t distribution corresponding to the significance level alpha of your choice (5%). The degrees of freedom (df) used in this test are :

df=nA+nB−2

Sp=Sqrt(9*0.6630 + 7* 0.72/10+8-2)=0.82

t=(677.98-574.24)/Sp* sqrt( 1/10+1/8)= -10.13 t-tabulated is as for 16 df and 0.05 level as 2.12

|t_cal| >t_tab, reject the null hypothesis . we here we reject the null hypothesis and conclude that A and B are diffreant from each other.

b) Conduct a hypothesis test to see if the variance in fish length is different in the Red River population compared to the variance in the Coldwater population.

Answer :

F_cal = S²_A / S²_B = 0.66/0.72 = ​​​​0.91

F_tab is as for the 9 and 7 df is 3.68,

F_cal> F_tab, then reject H0;

here we fail to reject the ho and conclude that variances are not equal.