Answer:
The margin of error for 90% confidence interval is 2.98, for 95% confidence interval is 3.55 and for 99% confidence interval is 4.68.
Step-by-step explanation:
The sample size is, <em>n</em> = 400.
The maximum value is, <em>Max.</em> = $225
The minimum value is, <em>Min</em>. = $80
The standard deviation of a distribution using the range of the data is:
![SD=\frac{Range}{4}=\frac{Max.-Min.}{4}=\frac{225-80}{4}= 36.25](https://tex.z-dn.net/?f=SD%3D%5Cfrac%7BRange%7D%7B4%7D%3D%5Cfrac%7BMax.-Min.%7D%7B4%7D%3D%5Cfrac%7B225-80%7D%7B4%7D%3D%2036.25)
As the sample is large, i.e. <em>n</em> = 400 > 30, according to the central limit theorem the sampling distribution of sample mean will follow a normal distribution.
The formula to compute the margin of error is:
![MOE=z_{\alpha /2}\times \frac{SD}{\sqrt{n}}](https://tex.z-dn.net/?f=MOE%3Dz_%7B%5Calpha%20%2F2%7D%5Ctimes%20%5Cfrac%7BSD%7D%7B%5Csqrt%7Bn%7D%7D)
- For a 90% confidence interval:
The value of <em>α</em> is 1 - 0.90 = 0.10
The critical value is,
(Use the standard normal table)
The MOE is:
![MOE=z_{\alpha /2}\times \frac{SD}{\sqrt{n}}\\=1.645\times \frac{36.25}{\sqrt{400}} \\=2.98](https://tex.z-dn.net/?f=MOE%3Dz_%7B%5Calpha%20%2F2%7D%5Ctimes%20%5Cfrac%7BSD%7D%7B%5Csqrt%7Bn%7D%7D%5C%5C%3D1.645%5Ctimes%20%5Cfrac%7B36.25%7D%7B%5Csqrt%7B400%7D%7D%20%5C%5C%3D2.98)
- For a 95% confidence interval:
The value of <em>α</em> is 1 - 0.95 = 0.05
The critical value is,
(Use the standard normal table)
The MOE is:
![MOE=z_{\alpha /2}\times \frac{SD}{\sqrt{n}}\\=1.96\times \frac{36.25}{\sqrt{400}} \\=3.55](https://tex.z-dn.net/?f=MOE%3Dz_%7B%5Calpha%20%2F2%7D%5Ctimes%20%5Cfrac%7BSD%7D%7B%5Csqrt%7Bn%7D%7D%5C%5C%3D1.96%5Ctimes%20%5Cfrac%7B36.25%7D%7B%5Csqrt%7B400%7D%7D%20%5C%5C%3D3.55)
- For a 99% confidence interval:
The value of <em>α</em> is 1 - 0.99 = 0.01
The critical value is,
(Use the standard normal table)
The MOE is:
![MOE=z_{\alpha /2}\times \frac{SD}{\sqrt{n}}\\=2.58\times \frac{36.25}{\sqrt{400}} \\=4.68](https://tex.z-dn.net/?f=MOE%3Dz_%7B%5Calpha%20%2F2%7D%5Ctimes%20%5Cfrac%7BSD%7D%7B%5Csqrt%7Bn%7D%7D%5C%5C%3D2.58%5Ctimes%20%5Cfrac%7B36.25%7D%7B%5Csqrt%7B400%7D%7D%20%5C%5C%3D4.68)
Thus, the margin of error for 90% confidence interval is 2.98, for 95% confidence interval is 3.55 and for 99% confidence interval is 4.68.