Answer:
a
[tex]0.716 < p < 0.744[/tex]
b
[tex]0.3498 < p < 0.4089[/tex]
c
With the result obtained from a and b the manager can be 95 % confidence that the proportion of the population that complained about dirty or ill-equipped bathrooms are within the interval obtained at a
and that
the proportion of the population that complained about loud or distracting diners at other tables are within the interval obtained at b
Step-by-step explanation:
From the question we are told that
The sample size is [tex]n = 1003[/tex]
The number that complained about dirty or ill-equipped bathrooms is [tex]e = 732[/tex]
The number that complained about loud or distracting diners at other tables is [tex]q = 381[/tex]
Given that the the confidence level is 95% then the level of significance is mathematically represented as
[tex]\alpha = (100- 95)\%[/tex]
[tex]\alpha = 0.05[/tex]
Next we obtain the critical value of [tex]\frac{\alpha }{2}[/tex] from the normal distribution table , the value is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
Considering question a
The sample proportion is mathematically represented as
[tex]\r p = \frac{e}{n}[/tex]
=> [tex]\r p = \frac{732}{1003}[/tex]
=> [tex]\r p = 0.73[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \sqrt{ \frac{ \r p (1- \r p)}{n} }[/tex]
[tex]E = 1.96* \sqrt{ \frac{ 0.73 (1- 0.73)}{1003} }[/tex]
[tex]E = 0.01402[/tex]
The 95% confidence interval is
[tex]\r p - E < p < \r p +E[/tex]
[tex]0.73 - 0.01402 < p < 0.73 + 0.01402[/tex]
[tex]0.716 < p < 0.744[/tex]
Considering question b
The sample proportion is mathematically represented as
[tex]\r p = \frac{q}{n}[/tex]
=> [tex]\r p = \frac{381}{1003}[/tex]
=> [tex]\r p = 0.3799[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \sqrt{ \frac{ \r p (1- \r p)}{n} }[/tex]
[tex]E = 1.96* \sqrt{ \frac{ 0.3799 (1- 0.3799)}{1003} }[/tex]
[tex]E = 0.0300[/tex]
The 95% confidence interval is
[tex]\r p - E < p < \r p +E[/tex]
[tex]0.3798 - 0.0300 < p < 0.3798 + 0.0300[/tex]
[tex]0.3498 < p < 0.4089[/tex]
