Are Nuclear Plants Safe? In a Gallup poll of 1038 adults, 47% said that nuclear plants are safe, 49% said that they are not safe, and 4% had no opinion. Construct a 95% confidence interval estimate of the proportion of all adults who say that nuclear plants are safe. Based on the result, can we conclude that fewer than half of all adults say that nuclear plants are safe?

Verified step by step guidance

1. Step 1: Identify the sample proportion (p̂) and sample size (n). From the problem, 47% of the 1038 adults said that nuclear plants are safe. Thus, p̂ = 0.47 and n = 1038.
2. Step 2: Determine the critical value (z*) for a 95% confidence level. For a 95% confidence interval, the critical value z* corresponds to the standard normal distribution and is approximately 1.96.
3. Step 3: Calculate the standard error (SE) of the sample proportion using the formula: SE = sqrt((p̂ * (1 - p̂)) / n). Substitute p̂ = 0.47 and n = 1038 into the formula.
4. Step 4: Compute the margin of error (ME) using the formula: ME = z* * SE. Use the z* value from Step 2 and the SE calculated in Step 3.
5. Step 5: Construct the confidence interval using the formula: Confidence Interval = p̂ ± ME. Interpret the interval to determine if fewer than half of all adults say that nuclear plants are safe by checking if the upper bound of the interval is less than 0.5.

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 with a specified level of confidence, typically 95%. It provides an estimate of uncertainty around a sample proportion, allowing researchers to infer about the population. For example, if a confidence interval for the proportion of adults who believe nuclear plants are safe is calculated, it helps assess whether this proportion is significantly different from 0.5.

Proportion

Proportion refers to the fraction of a whole, often expressed as a percentage, representing a specific characteristic within a population. In this context, it indicates the percentage of adults who believe nuclear plants are safe. Understanding how to calculate and interpret proportions is essential for analyzing survey data and making inferences about public opinion.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. In this scenario, it involves testing the null hypothesis that the proportion of adults who believe nuclear plants are safe is 0.5 against the alternative hypothesis that it is less than 0.5. This process helps determine if the sample results provide enough evidence to conclude that fewer than half of all adults hold this belief.