Question

In two consecutive years, 1000 adults in the magical community were asked if they believed that Voldemort had returned. The first year, 43% believed that he had returned and in the second year, 47% believed that he had returned.

1) Explain why it would be inappropriate to conclude, based on these percentages alone, that the percentage of adults who believed that Voldemort had returned increased from the first year to the second year.

2) Assume that the conditions have been satisfied. Construct a 95% confidence interval for the difference in the proportions of adults who believed that Voldemort had returned.

3) Based on the confidence interval, can we conclude that the proportion of adults with this belief increased from the first year to the second year? Explain.

Answer 1

a)

it would be inappropriate to conclude, based on these percentages alone, that the percentage of adults who believed that Voldemort had returned increased from the first year to the second year.

as the samples of people were taken from the magical community not from the adults

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b)

proportion success of sample 1 , p̂1=   x1/n1=   0.4300
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.470
level of significance, α =   0.05              
Z critical value =   Z α/2 =    1.960   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.0222          
margin of error , E = Z*SE =    1.960   *   0.0222   =   0.0436
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    -0.040   -   0.0436   =   -0.0836
upper limit = (p̂1 - p̂2) + E =    -0.040   +   0.0436   =   0.0036
                  
so, confidence interval is (   -0.0836   < p1 - p2 <   0.0036   )  

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c)

CI contain 0 ,so

we can conclude that the proportion of adults with this belief increased from the first year to the second year

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Please revert back in case of any doubt.

Please upvote. Thanks in advance.