Question

For each of the following sets of sample outcomes, construct the \(99 \%\) confidence interval for estimating \(P_{u}\). a. \(\begin{aligned} P_{s} &=0.40 \\ N &=548 \end{aligned}\) b. \(\begin{aligned} P_{s} &=0.37 \\ N &=522 \end{aligned}\) c. \(\begin{aligned} P_{s} &=0.79 \\ N &=121 \end{aligned}\) d. \(\begin{aligned} P_{s} &=0.14 \\ N &=100 \end{aligned}\) e. \(\begin{aligned} P_{s} &=0.43 \\ N &=1049 \end{aligned}\) f. \(\begin{aligned} P_{s} &=0.63 \\ N &=300 \end{aligned}\)

Short answer

a) (0.346, 0.454) b) (0.317, 0.423) c) (0.696, 0.884) d) (0.072, 0.208) e) (0.391, 0.469) f) (0.558, 0.702)

Step-by-step solution

- Understand the formula for confidence interval

The formula for the confidence interval for a population proportion \(\text{P}_{u}\) using a sample proportion \(\text{P}_{s}\) is given by: \[ \text{CI} = \text{P}_{s} \pm Z_{\alpha/2} \sqrt{ \frac{ \text{P}_{s}(1-\text{P}_{s}) } {N} }\] where \(Z_{\alpha/2}\) is the z-score corresponding to the desired confidence level, and \(N\) is the sample size.

- Determine the Z-score for a 99% confidence level

For a 99% confidence interval, \(Z_{\alpha/2} = 2.576\). This value is obtained from standard normal distribution tables.

- Apply the formula to each set of values

Calculate the confidence interval for each part by substituting the given \( \text{P}_{s} \) and \( N \) values into the confidence interval formula.

- Part a

\(\text{P}_{s} = 0.40\), \(N = 548\) Plugging in the values, the confidence interval is: \[ \text{CI} = 0.40 \pm 2.576 \sqrt{ \frac{0.40(1-0.40)}{548}} \approx 0.40 \pm 0.054 \Rightarrow (0.346, 0.454)\]

- Part b

\(\text{P}_{s} = 0.37\), \(N = 522\) Plugging in the values, the confidence interval is: \[ \text{CI} = 0.37 \pm 2.576 \sqrt{ \frac{0.37(1-0.37)}{522}} \approx 0.37 \pm 0.053 \Rightarrow (0.317, 0.423)\]

- Part c

\(\text{P}_{s} = 0.79\), \(N = 121\) Plugging in the values, the confidence interval is: \[ \text{CI} = 0.79 \pm 2.576 \sqrt{ \frac{0.79(1-0.79)}{121}} \approx 0.79 \pm 0.094 \Rightarrow (0.696, 0.884)\]

- Part d

\(\text{P}_{s} = 0.14\), \(N = 100\) Plugging in the values, the confidence interval is: \[ \text{CI} = 0.14 \pm 2.576 \sqrt{ \frac{0.14(1-0.14)}{100}} \approx 0.14 \pm 0.068 \Rightarrow (0.072, 0.208)\]

- Part e

\(\text{P}_{s} = 0.43\), \(N = 1049\) Plugging in the values, the confidence interval is: \[ \text{CI} = 0.43 \pm 2.576 \sqrt{ \frac{0.43(1-0.43)}{1049}} \approx 0.43 \pm 0.039 \Rightarrow (0.391, 0.469)\]

- Part f

\(\text{P}_{s} = 0.63\), \(N = 300\) Plugging in the values, the confidence interval is: \[ \text{CI} = 0.63 \pm 2.576 \sqrt{ \frac{0.63(1-0.63)}{300}} \approx 0.63 \pm 0.072 \Rightarrow (0.558, 0.702)\]

Key concepts

confidence interval

A confidence interval provides a range of values that is likely to contain a population parameter with a certain level of confidence. For example, a 99% confidence interval suggests that if we were to take 100 different samples and compute the interval for each sample, 99 of those intervals would contain the population parameter. The formula for the confidence interval for a population proportion (P_u) using a sample proportion (P_s) is:

\[ \text{CI} = \text{P}_{s} \pm Z_{\alpha/2} \sqrt{ \text{P}_{s}(1-\text{P}_{s}) }/N \]
Here, \( Z_{\alpha/2} \) is the z-score corresponding to the desired confidence level, and \( N \) is the sample size. This formula estimates the range within which the true population proportion is expected to be, based on the observed sample.

population proportion

The population proportion (P_u) is the fraction of the population that has a particular characteristic. For example, if we are interested in the proportion of people that support a particular policy in a town, the population proportion would be the actual fraction of all residents that support the policy. When we can't survey the entire population, we often use a sample proportion (P_s) which is the fraction of the sampled population that has the characteristic. For example, if 40% of a sample of 548 individuals support the policy, then \(P_s = 0.40\).

sample size

Sample size (N) is the number of observations or data points in the sample. It is a crucial factor in statistical analysis. Larger sample sizes generally lead to more precise estimates of the population parameter because they reduce the margin of error in the confidence interval. In our problems, the sample sizes vary, like 548 in part (a) and 100 in part (d). A larger N will typically produce a narrower confidence interval, indicating a more precise estimate.

z-score

The z-score (\(Z_{\alpha/2}\)) represents the number of standard deviations a data point is from the mean in a standard normal distribution. For a 99% confidence level, the z-score is 2.576. This means that we expect the true population proportion to fall within 2.576 standard deviations from the sample proportion 99% of the time. The z-score is a critical value that helps us determine the range or margin of error in our confidence interval.

standard normal distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is a symmetrical, bell-shaped distribution that allows us to use z-scores to find probabilities and critical values. Most statistical tables and calculations, including our confidence interval calculations, use the standard normal distribution to determine z-scores. It simplifies the process of finding the margin of error for different confidence levels.