Ellery Surveyed A Random Sample Of 25 Students From Her School. One Of The Questions In The Survey Required Students To State Their GPA Aloud. Based On The Responses Ellery Said She Was 90% Confident That The Interval From 0.40to 0.72 Captures The
Introduction
In statistics, a confidence interval is a range of values within which a population parameter is likely to lie. It is a crucial concept in statistical inference, allowing us to make conclusions about a population based on a sample of data. In this article, we will explore the concept of confidence intervals and examine Ellery's claim about the interval from 0.40 to 0.72 capturing the true mean GPA of her school's students.
What is a Confidence Interval?
A confidence interval is a range of values that is likely to contain the true population parameter. It is calculated from a sample of data and is used to estimate the population parameter with a certain level of confidence. The confidence level is usually expressed as a percentage, and it represents the degree of certainty that the interval contains the true population parameter.
Types of Confidence Intervals
There are two main types of confidence intervals: parametric and non-parametric. Parametric confidence intervals are used when the population distribution is known, and the sample size is large enough to assume that the sample is representative of the population. Non-parametric confidence intervals, on the other hand, are used when the population distribution is unknown, and the sample size is small.
Calculating a Confidence Interval
To calculate a confidence interval, we need to know the sample mean, the sample standard deviation, and the sample size. We also need to choose a confidence level, which is usually expressed as a percentage. The formula for calculating a confidence interval is:
CI = x̄ ± (Z * (σ / √n))
where:
- CI is the confidence interval
- x̄ is the sample mean
- Z is the Z-score corresponding to the desired confidence level
- σ is the sample standard deviation
- n is the sample size
Ellery's Claim
Ellery claims that she is 90% confident that the interval from 0.40 to 0.72 captures the true mean GPA of her school's students. To examine this claim, we need to know the sample mean, the sample standard deviation, and the sample size. Let's assume that the sample mean is 0.56, the sample standard deviation is 0.12, and the sample size is 25.
Calculating the Confidence Interval
Using the formula above, we can calculate the confidence interval as follows:
CI = 0.56 ± (1.645 * (0.12 / √25)) CI = 0.56 ± (1.645 * 0.024) CI = 0.56 ± 0.0396 CI = (0.5204, 0.5996)
Interpretation
The confidence interval (0.5204, 0.5996) is wider than Ellery's claim (0.40 to 0.72). This means that we are not 90% confident that the true mean GPA is within the interval from 0.40 to 0.72. In fact, the confidence interval suggests that the true mean GPA is likely to be between 0.52 and 0.60.
Conclusion
In conclusion, Ellery's claim that she is 90% confident that the interval from 0.40 to 0.72 captures the true mean GPA of her school's students is not supported by the data. The confidence interval suggests that the true mean GPA is likely to be between 0.52 and 0.60. This highlights the importance of understanding confidence intervals and interpreting statistical results correctly.
Understanding Confidence Intervals in Practice
Confidence intervals are a crucial concept in statistics, and they have many practical applications. Here are a few examples:
- Medical research: Confidence intervals are used to estimate the effectiveness of a new treatment or medication.
- Business: Confidence intervals are used to estimate the average cost of a product or service.
- Social sciences: Confidence intervals are used to estimate the average score on a test or survey.
Common Misconceptions about Confidence Intervals
There are several common misconceptions about confidence intervals that can lead to incorrect interpretations. Here are a few examples:
- Misconception 1: A confidence interval of 95% means that there is a 95% chance that the true population parameter is within the interval.
- Misconception 2: A confidence interval of 95% means that there is a 5% chance that the true population parameter is outside the interval.
- Misconception 3: A confidence interval of 95% means that the true population parameter is likely to be within the interval.
Conclusion
Q: What is a confidence interval?
A: A confidence interval is a range of values within which a population parameter is likely to lie. It is a crucial concept in statistical inference, allowing us to make conclusions about a population based on a sample of data.
Q: What is the difference between a confidence interval and a margin of error?
A: A confidence interval and a margin of error are related but distinct concepts. A confidence interval is a range of values within which a population parameter is likely to lie, while a margin of error is the maximum amount by which the sample estimate may differ from the true population parameter.
Q: How is a confidence interval calculated?
A: A confidence interval is calculated using the following formula:
CI = x̄ ± (Z * (σ / √n))
where:
- CI is the confidence interval
- x̄ is the sample mean
- Z is the Z-score corresponding to the desired confidence level
- σ is the sample standard deviation
- n is the sample size
Q: What is the significance of the confidence level?
A: The confidence level is the degree of certainty that the interval contains the true population parameter. A higher confidence level means that we are more confident that the interval contains the true population parameter.
Q: What is the difference between a 95% confidence interval and a 99% confidence interval?
A: A 95% confidence interval means that we are 95% confident that the interval contains the true population parameter, while a 99% confidence interval means that we are 99% confident that the interval contains the true population parameter.
Q: Can I use a confidence interval to make predictions about the future?
A: No, a confidence interval is not a prediction interval. It is a range of values within which a population parameter is likely to lie, based on the sample data.
Q: Can I use a confidence interval to compare two groups?
A: No, a confidence interval is not a test of significance. It is a range of values within which a population parameter is likely to lie, based on the sample data.
Q: What is the difference between a confidence interval and a prediction interval?
A: A confidence interval is a range of values within which a population parameter is likely to lie, based on the sample data, while a prediction interval is a range of values within which a future observation is likely to lie.
Q: Can I use a confidence interval to estimate the population mean?
A: Yes, a confidence interval can be used to estimate the population mean.
Q: Can I use a confidence interval to estimate the population proportion?
A: Yes, a confidence interval can be used to estimate the population proportion.
Q: What is the importance of understanding confidence intervals?
A: Understanding confidence intervals is crucial in statistics, as it allows us to make conclusions about a population based on a sample of data. It is also essential in making informed decisions in various fields, such as medicine, business, and social sciences.
Q: Can I use a confidence interval to compare two or more groups?
A: No, a confidence interval is not a test of significance. It is a range of values within which a population parameter is likely to lie, based on the sample data. However, you can use a confidence interval to compare two or more groups by calculating the confidence interval for each group and comparing the intervals.
Q: Can I use a confidence interval to estimate the population standard deviation?
A: Yes, a confidence interval can be used to estimate the population standard deviation.
Q: What is the difference between a confidence interval and a tolerance interval?
A: A confidence interval is a range of values within which a population parameter is likely to lie, based on the sample data, while a tolerance interval is a range of values within which a certain percentage of the population is likely to lie.
Q: Can I use a confidence interval to estimate the population variance?
A: Yes, a confidence interval can be used to estimate the population variance.
Q: What is the importance of choosing the correct confidence level?
A: Choosing the correct confidence level is crucial in statistics, as it affects the width of the confidence interval and the degree of certainty that the interval contains the true population parameter.
Q: Can I use a confidence interval to compare two or more proportions?
A: No, a confidence interval is not a test of significance. It is a range of values within which a population parameter is likely to lie, based on the sample data. However, you can use a confidence interval to compare two or more proportions by calculating the confidence interval for each proportion and comparing the intervals.
Q: Can I use a confidence interval to estimate the population correlation coefficient?
A: Yes, a confidence interval can be used to estimate the population correlation coefficient.
Q: What is the difference between a confidence interval and a prediction interval for a regression model?
A: A confidence interval for a regression model is a range of values within which the true population parameter is likely to lie, based on the sample data, while a prediction interval for a regression model is a range of values within which a future observation is likely to lie.