1. what is the margin of error calculated?\n2. what is the relationship between standard deviation and…

Explanation:

Step1: Define margin of error

The margin of error is a measure of the amount of random sampling error in a survey's results. It is calculated using the formula $E = z\times\frac{\sigma}{\sqrt{n}}$ (where $E$ is margin of error, $z$ is the z - score corresponding to the desired confidence level, $\sigma$ is the standard deviation and $n$ is the sample size).

Step2: Relationship between standard deviation and margin of error

From the formula $E = z\times\frac{\sigma}{\sqrt{n}}$, we can see that the margin of error is directly proportional to the standard deviation. When the standard deviation $\sigma$ increases, the margin of error $E$ increases, assuming $z$ and $n$ are constant.

Step3: Relationship between sample size and margin of error

The margin of error formula $E = z\times\frac{\sigma}{\sqrt{n}}$ shows an inverse - square root relationship between the sample size $n$ and the margin of error $E$. As the sample size $n$ increases, the margin of error $E$ decreases because we are dividing by a larger number under the square - root.

Step4: Answer part 4a

Eli has a sample size of $n_1 = 200$ and Elizabeth has a sample size of $n_2=2000$. Since the margin of error $E = z\times\frac{\sigma}{\sqrt{n}}$ and assuming the same population standard deviation $\sigma$ and confidence level (so the same $z$ - value), the sample with the smaller size will have a larger margin of error. So Eli will have the larger margin of error because his sample size is smaller.

Step5: Answer part 4b

We cannot determine who will have the larger proportion of residents supporting candidate A. The sample size only affects the margin of error and the precision of the estimate, not the actual proportion of support for a candidate in the sample. The proportion of residents supporting candidate A in each sample depends on the random selection of individuals in the samples and the true distribution of support in the population.

Step6: Find range for population mean

The confidence interval for the population mean $\mu$ is given by $\bar{x}-E\leq\mu\leq\bar{x} + E$, where $\bar{x}$ is the sample mean and $E$ is the margin of error. Given $\bar{x}=165$ cm and $E = 21$ cm, the range is $(165 - 21)\text{ cm}\leq\mu\leq(165 + 21)\text{ cm}$, so $144$ cm$\leq\mu\leq186$ cm.

Answer:

1. The margin of error is a measure of sampling error in survey results.
2. Margin of error is directly proportional to standard deviation.
3. Margin of error is inversely - related (inverse square - root relationship) to sample size.
4. a. Eli will have the larger margin of error because his sample size (200) is smaller than Elizabeth's (2000). b. We cannot determine who will have the larger proportion of residents supporting candidate A as sample size affects precision but not the actual proportion in the sample.
5. The range of values for the population mean is $144$ cm to $186$ cm.