One-Sided Confidence Interval A one-sided claim about a population proportion is a claim that the proportion is less than (or greater than) some specific value. Such a claim can be formally addressed using a one-sided confidence interval for p, which can be expressed as p<p+E or p>p-E, where the margin of error E is modified by replacing za/2 with za. (Instead of dividing between two tails of the standard normal distribution, put all of it in one tail.) The Chapter Problem refers to a Sallie Mae survey of 950 undergraduate students, and 53% of the survey subjects take online courses. Use that data to construct a one-sided 95% confidence interval that would be suitable for helping to determine whether the percentage of all undergraduates who take online courses is greater than 50%.

Verified step by step guidance

1. Step 1: Identify the given values from the problem. The sample size (n) is 950, the sample proportion (p̂) is 0.53 (53%), and the confidence level is 95%. For a one-sided confidence interval, the z-value (zα) corresponding to a 95% confidence level is approximately 1.645 (since all of the significance level is in one tail).
2. Step 2: Calculate the standard error (SE) of the sample proportion using the formula: SE = sqrt((p̂ * (1 - p̂)) / n). Substitute the values of p̂ = 0.53 and n = 950 into the formula.
3. Step 3: Compute the margin of error (E) for the one-sided confidence interval using the formula: E = zα * SE. Use the z-value of 1.645 and the standard error calculated in Step 2.
4. Step 4: Construct the one-sided confidence interval. Since the problem asks whether the percentage of all undergraduates who take online courses is greater than 50%, the one-sided confidence interval will be expressed as p > p̂ - E. Substitute the values of p̂ and E into this inequality.
5. Step 5: Interpret the result. The one-sided confidence interval provides a range of values for the population proportion p. If the lower bound of the interval is greater than 0.50, it supports the claim that the percentage of all undergraduates who take online courses is greater than 50%.

Key Concepts

One-Sided Confidence Interval

A one-sided confidence interval estimates a population parameter, such as a proportion, with a focus on one direction (either greater than or less than a specific value). Unlike two-sided intervals, which consider both tails of the distribution, one-sided intervals allocate the entire confidence level to one tail, allowing for a more precise assessment of claims about the population. This is particularly useful when testing hypotheses that predict a specific direction of effect.

Margin of Error

The margin of error quantifies the uncertainty associated with a sample estimate of a population parameter. It reflects the range within which the true population parameter is expected to lie, given a certain confidence level. In the context of one-sided confidence intervals, the margin of error is adjusted to account for the directional nature of the hypothesis, impacting how we interpret the results and make decisions based on the data.

Population Proportion

The population proportion is the fraction of a population that possesses a certain characteristic, often denoted as 'p'. In the context of surveys, it represents the percentage of respondents who affirmatively answer a question. Understanding the population proportion is crucial for constructing confidence intervals, as it serves as the basis for estimating how representative the sample data is of the larger population, influencing the conclusions drawn from statistical analyses.