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Magazine Chapter 24: Problem 11
A \(90 \%\) confidence interval for the mean score on the creativity test for those subjects who did not cheat is a. \(2.33 \pm 0.09\) b. \(2.33 \pm 0.14\) c. \(2.33 \pm 1.68\) d. \(1.00 \pm 0.14\).
Short Answer
Expert verified
Option (b) \(2.33 \pm 0.14\) is the correct 90% confidence interval.
Step by step solution
01
Understanding Confidence Interval
A confidence interval is a range of values, derived from the sample mean, that is used to estimate the population mean. It consists of a point estimate (the sample mean) and a margin of error.
02
Identifying the Correct Form
The format of the confidence interval in the choices is given as \( \, \text{mean} \pm \text{margin of error} \.\). Our goal is to find the option with the correct mean and margin of error for a 90% confidence level.
03
Evaluating the Mean
In the problem, the mean score identified for those who did not cheat is \(2.33\.\). Each given option should have a mean of \( 2.33 \, \) to be considered.
04
Checking the Margin of Error for 90% Confidence
For a 90% confidence interval, the margin of error should be small but significant to maintain accuracy without being too wide, typically much smaller than \(1.0 \.\) We will now evaluate which margin of error seems plausible among the options.
05
Analyzing Options
- Option (a): \(2.33 \pm 0.09\) seems reasonable for a typical study.- Option (b): \(2.33 \pm 0.14\) is slightly larger but still plausible.- Option (c): \(2.33 \pm 1.68\) is excessively wide and unlikely for a confidence interval.- Option (d): \(1.00 \pm 0.14\) wrong mean, therefore incorrect option.
06
Selecting the Most Plausible Option
The mean for the creativity test is \(2.33 \.\). Among options with a plausible margin of error for a 90% confidence interval, (b) \(2.33 \pm 0.14 \,\) is reasonable and matches the expected range.
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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 is a crucial component of any confidence interval. It represents the range within which we expect the true population parameter to fall, with a certain degree of confidence. This helps us understand the precision of our sample estimate.
- A smaller margin of error suggests more confidence in the sample estimate being close to the population mean.
- Conversely, a larger margin of error indicates greater uncertainty about how close the sample mean is to the population mean.
In our example, different options like \(2.33 \pm 0.09\) and \(2.33 \pm 0.14\) reflect varied margins of error. A choice like \(2.33 \pm 1.68\) is too wide, indicating less precision.
When constructing a confidence interval, it's important to choose a margin of error that balances accuracy with reliability.
- A smaller margin of error suggests more confidence in the sample estimate being close to the population mean.
- Conversely, a larger margin of error indicates greater uncertainty about how close the sample mean is to the population mean.
In our example, different options like \(2.33 \pm 0.09\) and \(2.33 \pm 0.14\) reflect varied margins of error. A choice like \(2.33 \pm 1.68\) is too wide, indicating less precision.
When constructing a confidence interval, it's important to choose a margin of error that balances accuracy with reliability.
Population Mean
The population mean is the average of a set of data points for an entire population. However, it's often impractical to measure every member of a population.
- Instead, we rely on the sample mean to estimate the population mean.
- The population mean provides the "true" average, but we use estimates due to resource and feasibility constraints.
In this problem, we are trying to estimate the population mean score on a creativity test for a group that did not cheat. Since obtaining the true population mean can be challenging, confidence intervals help provide a range in which the population mean is likely to lie, based on sampled data.
- Instead, we rely on the sample mean to estimate the population mean.
- The population mean provides the "true" average, but we use estimates due to resource and feasibility constraints.
In this problem, we are trying to estimate the population mean score on a creativity test for a group that did not cheat. Since obtaining the true population mean can be challenging, confidence intervals help provide a range in which the population mean is likely to lie, based on sampled data.
Sample Mean
The sample mean is the average of data collected from a subset of the population. It is a crucial part of inferential statistics because it gives us a practical measure to work with.
- The sample mean is used as the point estimate in constructing a confidence interval.
- Our confidence in the sample mean鈥檚 ability to represent the population mean is quantified using the confidence interval.
In the exercise provided, the mean score of \(2.33\) is calculated from the sample and used as a key point estimate. This sample mean serves as our best guess about what the true population mean might be.
- The sample mean is used as the point estimate in constructing a confidence interval.
- Our confidence in the sample mean鈥檚 ability to represent the population mean is quantified using the confidence interval.
In the exercise provided, the mean score of \(2.33\) is calculated from the sample and used as a key point estimate. This sample mean serves as our best guess about what the true population mean might be.
Confidence Level
The confidence level reflects the degree of certainty that the population mean lies within the confidence interval. It is expressed as a percentage.
- Common confidence levels are 90%, 95%, and 99%, with higher levels offering greater certainty.
- However, a higher confidence level also usually results in a wider margin of error.
In this exercise, a 90% confidence level was used. This means that, theoretically, if we were to take many samples and construct confidence intervals in the same way from each, 90% of those intervals would contain the true population mean.
Choosing an appropriate confidence level involves balancing the need for certainty with the desire for precision.
- Common confidence levels are 90%, 95%, and 99%, with higher levels offering greater certainty.
- However, a higher confidence level also usually results in a wider margin of error.
In this exercise, a 90% confidence level was used. This means that, theoretically, if we were to take many samples and construct confidence intervals in the same way from each, 90% of those intervals would contain the true population mean.
Choosing an appropriate confidence level involves balancing the need for certainty with the desire for precision.
