Answer :
Answer:
(1) The standard error is 0.0224.
(2) The error term is 0.058.
(3) The 99% confidence interval for the proportion of adults supporting the soft drink tax is (0.45, 0.57).
(4) Approximately 45% to 57% people support the soft drink tax.
Step-by-step explanation:
Let X = number of adults who support the soft drink tax.
The sample size is, n = 500.
The number of people who support the tax is, X = 255.
Compute the sample proportion as follows:
[tex]\hat p=\frac{X}{n}=\frac{255}{500}= 0.51[/tex]
(1)
Compute the standard error as follows:
[tex]SE_{\hat p}=\sqrt{\frac{\hat p(1-\hat p)}{n}} \\=\sqrt{\frac{0.51(1-0.51)}{500}}\\=0.0224[/tex]
Thus, the standard error is 0.0224.
(2)
The error term (Margin of error) is:
[tex]MOE=z_{\alpha/2}\times SE_{\hat p}[/tex]
For 99% confidence interval the critical value of z is:
[tex]z_{\alpha/2}=z_{0.01/2}=z_{0.005}=2.58[/tex]
Compute the value of MOE as follows:
[tex]MOE=z_{\alpha/2}\times SE_{\hat p}\\=2.58\times0.0224\\=0.057792\\\approx0.058[/tex]
Thus, the error term is 0.058.
(3)
Compute the 99% confidence interval as follows:
[tex]CI=\hat p\pm MOE\\=0.51\pm0.058\\=(0.452, 0.568)\\\approx(0.45, 0.57)[/tex]
Thus, the 99% confidence interval for the proportion of adults supporting the soft drink tax is (0.45, 0.57).
(4)
The report to be submitted to the mayor is that approximately 45% to 57% people support the soft drink tax.