Question

1) You wish to test the following claim (H1) at a significance level of α=0.05
      Ho:p1=p2
      H1:p1<p2
You obtain 23 successes in a sample of size n1=268 from the first population. You obtain 79 successes in a sample of size n2=465 from the second population. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution.
What is the critical value for this test? (Report answer accurate to three decimal places.)
critical value =
What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =

2) You wish to test the following claim (H1H1) at a significance level of α=0.005α=0.005.
      Ho:μ=70.9
      H1:μ≠70.9
You believe the population is normally distributed, but you do not know the standard deviation. You obtain the following sample of data:

data
71.8
55.5
105
28
46.2
37.1
58
51.4
52.4
53.5

What is the critical value for this test? (Report answer accurate to three decimal places.)
critical value = ±±
What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =

3) You wish to test the following claim (H1H1) at a significance level of α=0.05α=0.05.
      Ho:p1=p2
      H1:p1<p2
You obtain 23 successes in a sample of size n1=268n1=268 from the first population. You obtain 79 successes in a sample of size n2=465n2=465 from the second population. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution.
What is the critical value for this test? (Report answer accurate to three decimal places.)
critical value =
What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =

4) You wish to test the following claim (H1H1) at a significance level of α=0.10α=0.10.
      Ho:μ1=μ2
      H1:μ1<μ2
You believe both populations are normally distributed, but you do not know the standard deviations for either. However, you also have no reason to believe the variances of the two populations are not equal. You obtain the following two samples of data.

Sample #1	Sample #2	65.1 39 44.2 71.7 79.4 80.6 43.1 39 46.2 91.6 37.3 59.6 56.2 73.7 62.6 77.8 75.2 74.7 41.8 88.8 53.8 75.2 76.7 51.8 43.1 65.1	83.9 97.7 69.9 72.5 103.4 109.1 53 94 68.9 74.9 78.2 92.7 81.5 90.9 72.5

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =
What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value =

Answer 1

1) z-critical value , Z* =        -1.645 [Excel function =NORMSINV(α)  
------------

sample #1   ----->              
first sample size,     n1=   268          
number of successes, sample 1 =     x1=   23          
proportion success of sample 1 , p̂1=   x1/n1=   0.0858          
                  
sample #2   ----->              
second sample size,     n2 =    465          
number of successes, sample 2 =     x2 =    79          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.1699          
                  
difference in sample proportions, p̂1 - p̂2 =     0.0858   -   0.1699   =   -0.0841
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.1392          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.02654          
Z-statistic = (p̂1 - p̂2)/SE = (   -0.084   /   0.0265   ) =   -3.167
===================

2)

critical t value, t* =    ±   3.690
-----------

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   20.9029                  
Sample Size ,   n =    10                  
Sample Mean,    x̅ = ΣX/n =    55.8900                  
                          
degree of freedom=   DF=n-1=   9                  
                          
Standard Error , SE = s/√n =   20.9029   / √    10   =   6.6101      
t-test statistic= (x̅ - µ )/SE = (   55.890   -   70.9   ) /    6.6101   =   -2.271
==================

3)

z-critical value , Z* =        -1.645 [Excel function =NORMSINV(α)  
---------------

sample #1   ----->              
first sample size,     n1=   268          
number of successes, sample 1 =     x1=   23          
proportion success of sample 1 , p̂1=   x1/n1=   0.0858          
                  
sample #2   ----->              
second sample size,     n2 =    465          
number of successes, sample 2 =     x2 =    79          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.1699          
                  
difference in sample proportions, p̂1 - p̂2 =     0.0858   -   0.1699   =   -0.0841
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.1392          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.02654          
Z-statistic = (p̂1 - p̂2)/SE = (   -0.084   /   0.0265   ) =   -3.167