Question

estimating population proportions from samples:

1. if a sample proportion is 0.5 with a margin of error of 0.05, what is the plausible range of values for the population proportion? how do you know?
2. a survey of 100 students finds that 30 prefer basketball over soccer. what is the best estimate for the population proportion of students who prefer basketball?
3. when estimating a population mean from sample data, it is essential to include an associated ______________ with the sample mean. fill in the blank. then, explain in sentences the meaning of this sentence.
4. a sample proportion is 0.75 with a standard deviation of 0.025.

a. what is the margin of error?
b. what is the plausible range of values for the population proportion?

Explanation:

Step1: Recall margin - of - error formula for proportion range

The range of plausible values for a population proportion is given by $\text{Sample Proportion}\pm\text{Margin of Error}$.

Step2: Solve for problem 1 range

Given sample proportion $p = 0.5$ and margin of error $E=0.05$. The lower bound is $0.5 - 0.05=0.45$ and the upper bound is $0.5 + 0.05 = 0.55$. So the range is from $0.45$ to $0.55$.

Step3: Solve for problem 2 proportion

The best estimate for the population proportion $\hat{p}$ when given a sample is the sample proportion. Here, the sample proportion $\hat{p}=\frac{30}{100}=0.3$.

Step4: Fill in the blank for problem 3

When estimating a population mean from sample data, it is essential to include an associated margin of error with the sample mean. The margin of error represents the maximum expected difference between the sample mean and the true population mean. It gives a measure of the uncertainty associated with using a sample to estimate a population parameter.

Step5: Solve for problem 4a margin of error

For a proportion, if we assume a normal - approximation (for large samples), the margin of error $E$ is related to the standard deviation $\sigma$. For a sample proportion, when we don't have a confidence level specified and assume a simple relationship, the margin of error is approximately $E = 2\sigma$ (for a 95% - like confidence interval in a simple case). Given $\sigma=0.025$, then $E = 2\times0.025=0.05$.

Step6: Solve for problem 4b proportion range

Given sample proportion $p = 0.75$ and margin of error $E = 0.05$. The lower bound is $0.75-0.05 = 0.7$ and the upper bound is $0.75 + 0.05=0.8$. So the range is from $0.7$ to $0.8$.

Answer:

1. The plausible range of values for the population proportion is from $0.45$ to $0.55$.
2. The best estimate for the population proportion of students who prefer basketball is $0.3$.
3. Margin of error; The margin of error represents the maximum expected difference between the sample mean and the true population mean, indicating the uncertainty of the sample - based estimate.
4. a. The margin of error is $0.05$.

b. The plausible range of values for the population proportion is from $0.7$ to $0.8$.