Answer:
a
The 95% confidence interval is [tex]0.5753< p <0.62474[/tex]
b
This 95% confidence interval tell us that there 95% confidence that the true proportion of adults in the US that now believe life imprisonment is a better approach for punishing murder lies within the interval
c
The margin of error is [tex]E = 0.02474 [/tex]
d
The confidence interval becomes wider
Step-by-step explanation:
From the question we are told that
The sample size is n = 1506
The sample proportion is [tex]\^ p = 0.60[/tex]
From the question we are told the confidence level is 95% , hence the level of significance is
[tex]\alpha = (100 - 95 ) \%[/tex]
=> [tex]\alpha = 0.05[/tex]
Generally from the normal distribution table the critical value of [tex]\frac{\alpha }{2}[/tex] is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \sqrt{\frac{\^ p (1- \^ p)}{n} } [/tex]
=> [tex]E = 1.96 * \sqrt{\frac{0.60 (1- 0.60)}{1506} } [/tex]
=> [tex]E = 0.02474 [/tex]
Generally 95% confidence interval is mathematically represented as
[tex]\^ p -E < p < \^ p +E[/tex]
=> [tex]0.60 - 0.02474 < p <0.60 + 0.02474[/tex]
=> [tex]0.5753< p <0.62474[/tex]
This 95% confidence interval tell us that there 95% confidence that the true proportion of adults in the US that now believe life imprisonment is a better approach for punishing murder lies within the interval
Generally the level of confidence varies directly with the critical value of [tex]\frac{\alpha }{2}[/tex] and this in turn varies directly with the margin of error which when sample proportion is constant it determines the width of the confidence interval so when the level of confidence increases the confidence interval becomes wider
