Question

Provided below are summary statistics for independent simple random samples from two populations. Use the pooled​ t-test and the pooled​ t-interval procedure to conduct the required hypothesis test and obtain the specified confidence interval.

x1=15​,

s1=2.3​,

n1=15​,

x2=16​,

s2=2.2​,

n2=15

a.​ First, what are the correct hypotheses for a​ two-tailed test?

compute the test statistic.

Now determine the critical values.

What is the conclusion of the hypothesis​ test?

Since the test statistic ____ in the rejection​ region, ____ H0

b. The 95​% confidence interval is from?

Answer 1

a)

Ho :   µ1 - µ2 =   0
Ha :   µ1-µ2 ╪   0

Sample #1   ---->   1
mean of sample 1,    x̅1=   15.000
standard deviation of sample 1,   s1 =    2.300
size of sample 1,    n1=   15
      
Sample #2   ---->   2
mean of sample 2,    x̅2=   16.000
standard deviation of sample 2,   s2 =    2.200
size of sample 2,    n2=   15
      
difference in sample means = x̅1 - x̅2 =    15-16=   -1.0000
pooled std dev , Sp=   √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) =    2.2506
std error , SE =    Sp*√(1/n1+1/n2) =    0.8218
t-statistic = ((x̅1-x̅2)-µd)/SE =    (-1-0)/0.8218=   -1.2169
      
Level of Significance ,    α =    0.05
Degree of freedom, DF=   n1+n2-2 =    28
t-critical value , t* = ±   2.048   (excel formula =t.inv(α/2,df)
Decision:   | t-stat | < | critical value |, so, Do not Reject Ho  

Since the test statistic __do not fall__ in the rejection​ region, __fail to reject __ H0

b)

Degree of freedom, DF=   n1+n2-2 =    28
t-critical value = t α/2 =    2.0484   (excel formula =t.inv(α/2,df)
      
pooled std dev , Sp=   √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) =    2.2506
std error , SE =    Sp*√(1/n1+1/n2) =    0.8218
margin of error, E = t*SE =    2.048*0.8218=   1.6834
      
difference in sample means = x̅1 - x̅2 =    15-16=   -1.0000
confidence interval is       
Interval Lower Limit= (x̅1-x̅2) - E =    -1-1.6834=   -2.6834
Interval Upper Limit= (x̅1-x̅2) + E =    -1+1.6834=   0.6834