Problem 61 - Basic Statistics for Business and Economics with Student

Question

In a recent Zogby poll of 1,000 adults nationwide, 613 said they believe other forms of life exist elsewhere in the universe. Construct the 99 percent confidence interval for the population proportion of those believing life exists elsewhere in the universe. Does your result imply that a majority of Americans believe life exists outside of Earth?

Step-by-Step Solution

Answer

Yes, the confidence interval suggests a majority of Americans believe in extraterrestrial life.

Step 1: Define the problem and identify variables

We have a sample size of 1,000 adults, of which 613 believe that life exists elsewhere in the universe. We need to create a confidence interval for the population proportion \( p \) of those who believe there is life elsewhere. The sample proportion \( \hat{p} \) is calculated as \( \hat{p} = \frac{613}{1000} = 0.613 \). We want a 99% confidence interval.

Step 2: Determine the critical value

For a 99% confidence level, we use the standard normal distribution (Z-distribution). The critical value \( z \) for 99% confidence can be found in the Z-table: \( z \approx 2.576 \).

Step 3: Calculate the standard error

The standard error (SE) of the sample proportion is calculated using the formula: \( SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \), where \( n \) is the sample size. Thus, \( SE = \sqrt{\frac{0.613 \times (1 - 0.613)}{1000}} \approx 0.0154 \).

Step 4: Construct the confidence interval

The confidence interval is given by \( \hat{p} \pm (z \times SE) \). Substituting the values, \( 0.613 \pm (2.576 \times 0.0154) \). Calculating, \( 0.613 \pm 0.0397 \). The confidence interval is \( (0.5733, 0.6527) \).

Step 5: Interpret the confidence interval

The 99% confidence interval for the population proportion is \( (0.5733, 0.6527) \). Since the entire interval is above 0.5, it implies that the majority of Americans, with a confidence level of 99%, believe that life exists outside of Earth.

Key Concepts

Population Proportion

Understanding population proportion is important when dealing with surveys. It is a way to describe the fraction of the entire population that shares a specific trait or opinion.
The population proportion is denoted by the symbol \( p \). In this exercise, \( p \) represents the proportion of adults who believe that other life forms exist in the universe. Since we often can't ask the entire population, we use a sample.

• In the sample, we find \( \hat{p} \), which is our sample proportion.
• This sample proportion is calculated by dividing the number of individuals with the desired trait by the total sample size.

Using the provided example, \( \hat{p} = \frac{613}{1000} = 0.613 \), indicates that 61.3% of this sample believe in extraterrestrial life.
We use this sample proportion to make inferences about the entire population.

Sample Size

Sample size, represented as \( n \), is a key factor in how accurately the sample reflects the entire population. A larger sample size gives us more confidence in our predictions.

• It affects the margin of error and the precision of the estimated population parameter.
• When the sample size is big, the standard error decreases, leading to a narrower confidence interval.

In this exercise, the sample size is 1,000. This is considered a substantial sample for deriving a reliable estimate of the population proportion. Larger samples typically provide more information and reduce uncertainty.

Standard Error

The standard error (SE) is crucial in statistics, as it measures the variation or "spread" of a sampling distribution.

• The SE helps determine how much the sample proportion estimates vary from sample to sample.
• A smaller SE indicates that the sample proportion is a more accurate estimate of the population proportion \( p \).

The formula for calculating the standard error is: \( SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \).
In this example, using \( \hat{p} = 0.613 \) and \( n = 1000 \), we find \( SE \approx 0.0154 \). This tells us about the expected fluctuation in sample proportions if we repeatedly drew samples from the population.

Z-distribution

The Z-distribution, a type of normal distribution, is fundamental in calculating confidence intervals, especially when dealing with proportions.

• It is often called the standard normal distribution, with a mean of 0 and a standard deviation of 1.
• Critical values from the Z-distribution help determine the confidence interval for a proportion.
• For a 99% confidence interval, we look up the critical value of approximately 2.576. This value is used to extend the interval on either side of the sample proportion.

In the context of this exercise, the Z-distribution helps us understand the likelihood that the true population proportion lies within the computed interval from the sample proportion. By knowing the Z-value, we can establish how confident we are in our interval estimate.