Question

Suppose that a city planning commission wants to know the proportion of city residents who support installing streetlights in the downtown area. Two different people independently selected random samples of city residents and used their sample data to construct the following confidence intervals for the population proportion: Interval 1:(0.28,0.34) Interval 2:(0.31,0.33) (Hint: Consider the formula for the confidence interval given on page 401 ) a. Explain how it is possible that the two confidence intervals are not centered in the same place. b. Which of the two intervals conveys more precise information about the value of the population proportion? c. If both confidence intervals have a \(95 \%\) confidence level, which confidence interval was based on the smaller sample size? How can you tell? d. If both confidence intervals were based on the same sample size, which interval has the higher confidence level? How can you tell?

Short answer

a. The center of the intervals differ due to the difference in means of the independent samples. b. Interval 2 conveys more precise information as it is narrower. c. If both have a 95% confidence level, Interval 1 was likely based on a smaller sample size due to its width. d. If both intervals are based on the same sample size, Interval 2 would most likely have a higher confidence level due to its narrower width.

Step-by-step solution

Explaining the center of the intervals

The center of a confidence interval is not determined by the population but by the sample that is drawn. In this case, the independent samples most likely had different means, leading to differently centered confidence intervals.

Determining precision

The narrower interval provides more precise information about the population proportion. Interval 2 (0.31, 0.33) is narrower than Interval 1 (0.28, 0.34), so Interval 2 conveys more precise information.

Identifying Confidence Interval with a Smaller Sample Size

Wider confidence intervals are typically associated with smaller sample sizes. In this case, the wider interval is Interval 1 (0.28, 0.34). Therefore, if both confidence intervals have a 95% confidence level, Interval 1 was likely based on a smaller sample size.

Determining the interval with a higher Confidence Level

More narrow confidence intervals are representative of higher confidence levels. Therefore, if both intervals are based on the same sample size, Interval 2 (0.31, 0.33) would have a higher confidence level.

Key concepts

Population Proportion

Understanding the population proportion involves recognizing it as the true percentage or fraction of a population exhibiting a particular characteristic. For example, when the city planning commission desires to identify the proportion of residents supporting the installation of streetlights, they are seeking the population proportion.

However, gauging the actual population proportion can be cumbersome or impractical, necessitating the use of a sample to estimate the proportion. The sample's findings can then be scaled to infer the proportion for the broader population through a process called statistical inference. This inference comes with a level of uncertainty, reflected in the confidence intervals provided.

Sample Size

The sample size, which is the number of observations included in a sample, plays a crucial role in determining the precision and reliability of our estimate. A larger sample size generally reduces the margin of error and gives narrower confidence intervals, suggesting a more precise estimate of the population proportion.

When comparing the two intervals from the exercise, the larger width of Interval 1 indicates a likely smaller sample size was used compared to Interval 2. This is because a smaller sample size will produce a wider confidence interval, given the same confidence level, due to increased variability and uncertainty in the estimate.

Confidence Level

The confidence level is a measure of how often the calculated confidence interval would contain the true population parameter if you repeated the sample many times. A 95% confidence level, as mentioned in the exercise, implies that if we were to take 100 different samples and construct confidence intervals in the same way, we would expect about 95 of those intervals to contain the population proportion.

In practical terms, a higher confidence level means that we can be more assured that the interval includes the true proportion, but it also results in a wider interval. Conversely, a lower confidence level would give us a narrower interval, implying less certainty that the interval includes the true proportion.

Precision of Intervals

The precision of a confidence interval is linked to its width; a narrower interval denotes a more precise estimate of the population parameter. When Interval 2 is more narrow than Interval 1, it provides more exact information about the population proportion's value.

It's important to note that precision is a double-edged sword. While we desire precise estimates, precision inevitably comes with trade-offs, such as the need for larger sample sizes or acceptance of a lower confidence level. Precision in interval estimates is not just an indicator of quality, but also of the resources and risks entailed in the sampling process.