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 <em>X</em> = number of adults who support the soft drink tax.
The sample size is, <em>n</em> = 500.
The number of people who support the tax is, <em>X</em> = 255.
Compute the sample proportion as follows:
![\hat p=\frac{X}{n}=\frac{255}{500}= 0.51](https://tex.z-dn.net/?f=%5Chat%20p%3D%5Cfrac%7BX%7D%7Bn%7D%3D%5Cfrac%7B255%7D%7B500%7D%3D%200.51)
(1)
Compute the standard error as follows:
![SE_{\hat p}=\sqrt{\frac{\hat p(1-\hat p)}{n}} \\=\sqrt{\frac{0.51(1-0.51)}{500}}\\=0.0224](https://tex.z-dn.net/?f=SE_%7B%5Chat%20p%7D%3D%5Csqrt%7B%5Cfrac%7B%5Chat%20p%281-%5Chat%20p%29%7D%7Bn%7D%7D%20%5C%5C%3D%5Csqrt%7B%5Cfrac%7B0.51%281-0.51%29%7D%7B500%7D%7D%5C%5C%3D0.0224)
Thus, the standard error is 0.0224.
(2)
The error term (Margin of error) is:
![MOE=z_{\alpha/2}\times SE_{\hat p}](https://tex.z-dn.net/?f=MOE%3Dz_%7B%5Calpha%2F2%7D%5Ctimes%20SE_%7B%5Chat%20p%7D)
For 99% confidence interval the critical value of <em>z</em> is:
![z_{\alpha/2}=z_{0.01/2}=z_{0.005}=2.58](https://tex.z-dn.net/?f=z_%7B%5Calpha%2F2%7D%3Dz_%7B0.01%2F2%7D%3Dz_%7B0.005%7D%3D2.58)
Compute the value of MOE as follows:
![MOE=z_{\alpha/2}\times SE_{\hat p}\\=2.58\times0.0224\\=0.057792\\\approx0.058](https://tex.z-dn.net/?f=MOE%3Dz_%7B%5Calpha%2F2%7D%5Ctimes%20SE_%7B%5Chat%20p%7D%5C%5C%3D2.58%5Ctimes0.0224%5C%5C%3D0.057792%5C%5C%5Capprox0.058)
Thus, the error term is 0.058.
(3)
Compute the 99% confidence interval as follows:
![CI=\hat p\pm MOE\\=0.51\pm0.058\\=(0.452, 0.568)\\\approx(0.45, 0.57)](https://tex.z-dn.net/?f=CI%3D%5Chat%20p%5Cpm%20MOE%5C%5C%3D0.51%5Cpm0.058%5C%5C%3D%280.452%2C%200.568%29%5C%5C%5Capprox%280.45%2C%200.57%29)
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.