Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.


Job Interviews In a Harris poll of 514 human resource professionals, 45.9% said that body piercings and tattoos were big personal grooming red flags.


c. Repeat part (b) using a confidence level of 80%.

Verified step by step guidance

1. Step 1: Identify the given values from the problem. The sample size (n) is 514, and the sample proportion (p̂) is 45.9%, which can be written as p̂ = 0.459. The confidence level is 80%, so the corresponding z-score (critical value) for an 80% confidence level needs to be determined.
2. Step 2: Find the z-score for an 80% confidence level. To do this, calculate the area in each tail of the standard normal distribution. For an 80% confidence level, 100% - 80% = 20%, so each tail has 10% (0.10). Use a z-table or statistical software to find the z-score corresponding to a cumulative area of 0.90 (1 - 0.10).
3. Step 3: Calculate the standard error (SE) of the sample proportion using the formula: SE = sqrt((p̂ * (1 - p̂)) / n). Substitute the values of p̂ = 0.459 and n = 514 into the formula.
4. Step 4: Compute the margin of error (ME) using the formula: ME = z * SE, where z is the critical value obtained in Step 2 and SE is the standard error calculated in Step 3.
5. Step 5: Construct the confidence interval for the population proportion (p) using the formula: Confidence Interval = p̂ ± ME. Substitute the values of p̂ and ME to find the lower and upper bounds of the confidence interval.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed with a confidence level, such as 80%, indicating the probability that the interval will capture the true value if the experiment were repeated multiple times. For example, if a confidence interval for a proportion is calculated as (0.40, 0.50), we can be 80% confident that the true proportion lies within this range.

Sample Proportion

The sample proportion is the ratio of the number of successes in a sample to the total number of observations in that sample. In the context of the question, it refers to the percentage of human resource professionals who view body piercings and tattoos as grooming red flags. This value is crucial for calculating the confidence interval, as it serves as the point estimate for the population proportion.

Margin of Error

The margin of error quantifies the uncertainty associated with a sample estimate. It is calculated based on the sample size and the variability of the data, and it determines how far the sample proportion may deviate from the true population proportion. In constructing a confidence interval, the margin of error is added to and subtracted from the sample proportion to create the upper and lower bounds of the interval.