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Chapter 24: Problem 10

A \(95 \%\) confidence interval for the mean score on the creativity test for those subjects who cheated is (a) \(3.60 \pm 0.19\). (b) \(3.60 \pm 0.39\). (c) \(3.60 \pm 2.02\). (d) \(1.27 \pm 0.43\).

Short Answer

Expert verified
The correct answer is (a) \(3.60 \pm 0.19\).

Step by step solution

01

Understanding the Confidence Interval

A confidence interval is a range of values that is likely to contain a population parameter with a certain level of confidence, in this case, 95%. It is generally represented as \( \bar{x} \pm ME \), where \( \bar{x} \) is the sample mean and \( ME \) is the margin of error.
02

Identify the Given Mean

In the options, the format \( 3.60 \pm ME \) indicates that the mean \( \bar{x} \) is \( 3.60 \) for options (a), (b), and (c). Only in option (d) is the mean different, which is \( 1.27 \). We need to focus on the options with a mean of \( 3.60 \).
03

Look for a Reasonable Margin of Error

A 95% confidence interval uses a margin of error that depends on the data's standard deviation and sample size. Typically, a very large margin of error might not seem reasonable for a test score.
04

Analyze the Options

Examine the provided options:- Option (a): \( 3.60 \pm 0.19 \)- Option (b): \( 3.60 \pm 0.39 \)- Option (c): \( 3.60 \pm 2.02 \)- Option (d): \( 1.27 \pm 0.43 \)Typically, for test scores, a margin of error in the lower decimal range is reasonable. Options (b) and (c) seem too large given the context of test scores.
05

Choose the Correct Option

Given standard practices in statistics, the margin of error for a 95% confidence interval should generally not be too large. Therefore, \( 0.19 \) from option (a) is the most reasonable margin of error.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Margin of Error
The margin of error (ME) is a critical component in statistics, especially when constructing confidence intervals. It represents the range we expect the true population parameter to fall within our estimate, given a certain level of confidence. In simpler terms, it is the buffer zone on either side of a sample estimate.
When looking at a 95% confidence interval, the margin of error tells us how much we can expect the sample mean (or estimate) to differ from the actual population parameter.
  • It is calculated using the standard deviation, sample size, and the z-score or t-score corresponding to the desired level of confidence.
  • Typically, in a test score context, smaller values of margin of error like 0.19 indicate higher precision in estimating the population mean compared to larger values.
Thus, a smaller margin of error reflects that the sample mean is a more accurate estimator of the actual population parameter. This makes the confidence interval tighter and more reliable.
Sample Mean
The sample mean (\( \bar{x} \)) is what we compute to estimate the average of a particular set of observations. In statistical terms, it serves as a point estimate of the population mean, meaning it provides a single best guess.
The formula for the sample mean is simple: sum all the observed values and divide by the number of observations.
  • For example, in our exercise, the sample mean is given as 3.60 for most of the options, representing the average creativity score derived from the sample of subjects who cheated.
  • This value, along with the margin of error, helps us construct the 95% confidence interval.
  • The closer the sample mean is to the true population mean, the more reliable our confidence interval will be.
The role of the sample mean in statistics is foundational, serving as the basis from which predictions and inferences about the wider population are made.
Population Parameter
In the world of statistics, the population parameter represents the true value we aim to comprehend through our analysis. Unlike the sample statistics, which are calculated merely from sample data, population parameters are theoretical and usually unknown constants.
They include quantities such as the population mean or proportion that describe the entire dataset.
  • In our context, the term specifically refers to the average creativity scores of all those who cheated under certain conditions, not just the sampled subjects.
  • Statistics aims to use sample data鈥攍ike the sample mean we found鈥攖o infer or approximate these unknown population values accurately.
  • The accuracy of our approximation is then gauged by examining the confidence interval, which gives us a likely range where the true population parameter lies.
Understanding population parameters is crucial because they enable us to make informed claims about larger groups, even when measurement of every individual is impractical.

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Most popular questions from this chapter

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