Question

13. You wish to test the following claim (Ha) at a significance level of α=0.002.
  Ho:p1 = p2
      Ha:p1 > p2
You obtain 17 successes in a sample of size n1=387 from the first population. You obtain 15 successes in a sample of size n2=476 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 p-value for this sample? (Report answer accurate to four decimal places.)
p-value =

14. You wish to test the following claim (Ha) at a significance level of α=0.001.

      Ho:μ1 = μ2
      Ha: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 a sample of size n1=23 with a mean of M1=85.7 and a standard deviation of SD1=19.6 from the first population. You obtain a sample of size n2=19 with a mean of M2=71.7 and a standard deviation of SD2=15.4 from the second population.

What is the p-value for this test? For this calculation, use the conservative under-estimate for the degrees of freedom as mentioned in the textbook. (Report answer accurate to four decimal places.)
p-value =

Answer 1

13)

Ho:    p1 - p2 =    0
Ha:    p1 - p2 >    0
        
first sample size,                                      n1=    387
number of successes, sample 1 =        x1=    17
proportion success of sample 1 , p̂1=    x1/n1=    0.044
        
second sample size,                                      n2 =     476
number of successes, sample 2 =        x2 =     15
proportion success of sample 1 , p̂ 2=    x2/n2   =     0.031512605
        
difference in sample proportions,     p̂1 - p̂2 =      0.012415044
        
pooled proportion , p =    (x1+x2)/(n1+n2)=    0.0371
        
std error ,SE =     =SQRT(p*(1-p)*(1/n1+ 1/n2)=    0.0129
        
Z-statistic =     (p̂1 - p̂2)/SE =      0.9599
        
z-critical value , Z* =         1.6449
p-value =         0.1685    [excel functon:  =normsdist(-0.9599)

P-value >α, fail to reject Ho

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14)

Ho :    µ1 - µ2 =    0
Ha :    µ1-µ2 >    0
        
Level of Significance ,     α =     0.001
        
mean of sample 1,                   x̅1=    85.7
standard deviation of sample 1,    s1 =     19.6
size of sample 1,                         n1=    23
        
mean of sample 2,                   x̅2=    71.7
standard deviation of sample 2,    s2 =     15.4
size of sample 2,                         n2=    19
        
difference in sample means =     x̅1-x̅2   =     14.0000
        
std error , SE =     √(s1²/n1+s2²/n2) =     5.4023
        
t-statistic   =     ((x̅1-x̅2)-µd)/SE   =     2.5915

DF = min(n1-1 , n2-1 )=    18

p-value=0.0092

p-value>α , Do not reject null hypothesis