INTRODUCTION – Confidence Intervals
For the rest of the section, students will engage with the fundamental, but complicated, idea of the confidence interval. Students will also explore how data professionals use confidence intervals to indicate the uncertainty that accompanies an estimate. The student learns how to build confidence intervals, develop interpretations, and understand the potential misinterpreted aspects of the process.
Unlike the standard model examples, this presentation will involve real-world scenarios, making participants’ understanding of the theoretical foundations and application of this essential statistical inference much easier. Therefore, this thorough overview will prove most beneficial to anyone who wants to deepen their understanding of statistics and base their decisions on the ambiguity of estimation in data.
Learning Objectives:
- Use Python to create a confidence interval.
- Explain the construction of a confidence interval for means and proportions.
- Identify common misinterpretations related to confidence intervals.
- Interpret a confidence interval correctly.
- Define key terms related to confidence intervals, such as confidence level and margin of error.
- Differentiate between point estimates and interval estimates.
PRACTICE QUIZ: Test your knowledge on Analytical Thinking
1. Which of the following statements describes an interval estimate?
- An interval estimate uses a range of values to estimate a sample statistic.
- An interval estimate uses a range of values to estimate a population parameter. (CORRECT)
- An interval estimate uses a single value to estimate a population parameter.
- An interval estimate uses a single value to estimate a sample statistic.
Correct: With an interval estimate, rather than a precise single point value, you have a range of points that ensure that a population parameter will most likely fall within it. Such a range guarantees the uncertainty and variability inherent in the estimation process.
2. What is the maximum expected difference between a population parameter and a sample estimate?
- Confidence level
- Margin of error (CORRECT)
- Standard deviation
- Range
Correct: Ranging to some extent it is an experimental approximation difference with respect to the maximum population parameter difference that has to fall into any sample estimation. This experimental difference is nothing other than the expression of discrepancy or variability in estimating the true population parameter.
3. A 95% confidence interval means that 95% of all the data values in the dataset fall within the interval.
- False (CORRECT)
- True
Correct: A 95% confidence level would be equivalent to saying that the estimation process has succeeded 95% of the times making this kind of estimation. That means when repeating the sampling many times, 95% of all such confidence intervals would include the true population parameter. It does not mean that 95% of values in the dataset are contained in that interval. This is a significant misconception. The confidence interval tells you how precise your estimate is, not how scattered your data are.
PRACTICE QUIZ: TEST YOUR KNOWLEDGE: CONSTRUCT CONFIDENCE INTERVALS
1. After identifying a sample statistic, what is the proper order of the next three steps of constructing a confidence interval?
- Find the margin of error, calculate the interval, and choose a confidence level
- Choose a confidence level, calculate the interval, and find the margin of error.
- Choose a confidence level, find the margin of error, and calculate the interval (CORRECT)
- Find the margin of error, choose a confidence level, and calculate the interval
Correct: A confidence interval consists of identifying a sample statistic, then choosing a confidence level, establishing a margin of error, and calculating the interval.
2. A data professional is working for an online retail company. Their manager asks them to estimate the mean time customers spend on the company’s website. They construct a confidence interval based on a sample mean of 50 seconds and a margin of error of 4 seconds. What is the interval?
- [50, 54]
- [46, 54] (CORRECT)
- [46, 50]
- [54, 46]
Correct: This interval corresponds to 46 to 54. The lower limit is obtained by subtracting the margin of error from the sample mean: 50-4=46. The upper limit is obtained by adding the margin of error to the sample mean: 50+4=54.
3. What happens as a sample size gets larger? Select all that apply.
- The margin of error increases.
- The confidence interval widens.
- The confidence interval narrows. (CORRECT)
- The margin of error decreases. (CORRECT)
Correct: Another way that it works is that the confidence interval tends to get narrower as the sample size increases. With larger sample size, the margin of error shrinks and becomes zero at the maximum size, that is, if everybody of the population were sampled, then it would be’marginal error-free’ .
PRACTICE QUIZ: TEST YOUR KNOWLEDGE: WORK WITH CONFIDENCE INTERVALS IN PYTHON
1. What Python function enables a data professional to compute the standard deviation term in the sample standard error of a mean?
- pandas.DataFrame.std() (CORRECT)
- pandas.DataFrame.median()
- pandas.DataFrame.mode()
- pandas.DataFrame.hist()
Correct: Using the pandas.DataFrame.std() function, a data professional can calculate the standard deviation part of the sample standard error of the mean. The sample standard error can simply be calculated by dividing the sample standard deviation by the square root of the sample size.
2. A data professional is constructing a confidence interval of the sample mean using the function scipy.stats.norm.interval(). What arguments should they specify? Select all that apply.
- iqr, which they set to the interquartile range
- confidence (a.k.a. “alpha”), which they set to the confidence level (CORRECT)
- loc, which they set to the sample mean (CORRECT)
- scale, which they set to the sample standard error (CORRECT)
Correct: Confidence (alpha); loc (set to sample mean); and scale will be the sample standard error.
MODULE 4 CHALLENGE
1. What is a key difference between a point estimate and an interval estimate?
- A point estimate uses a single value to estimate a population parameter; an interval estimate uses a range of values to estimate a population parameter. (CORRECT)
- A point estimate uses a range of values to estimate a sample statistic; an interval estimate uses a single value to estimate a sample statistic.
- A point estimate uses a range of values to estimate a population parameter; an interval estimate uses a single value to estimate a population parameter.
- A point estimate uses a single value to estimate a sample statistic; an interval estimate uses a range of values to estimate a sample statistic.
2. A data professional working for a moving company is estimating the average time it takes to complete a move. Based on a sample mean of 3 hours, they construct the following 95% confidence interval: [2.5 , 3.5]. What does 95% refer to?
- Evaluating margin of error (CORRECT)
- Constructing a confidence level
- Defining a sample statistic
- Choosing a sampling distribution
3. A data professional working for a moving company is estimating the average time it takes to complete a move. Based on a sample mean of 3 hours, they construct the following 95% confidence interval: [2.5 , 3.5]. What does 95% refer to?
- The percentage of all possible sample means that fall within the range of the interval
- The success rate of the estimation process (CORRECT)
- The margin of error
- The percentage of data values in the dataset
4. A data analytics team with a clothing manufacturer constructs a confidence interval to help estimate future returns. First, they identify the sample statistic. Then, they choose a confidence level of 95%. According to the four steps to constructing a confidence interval for a proportion, what should they do next?
- Plot a histogram
- Choose a confidence level
- Calculate the interval
- Find the margin of error (CORRECT)
5. A data professional working for a light bulb manufacturer is estimating the mean bulb lifespan based on sample data. They construct a 95% confidence interval using a sample size of 100. In addition, they construct a 95% confidence interval using a sample size of 1,000. What happens as the sample size increases?
- The margin of error decreases. (CORRECT)
- The margin of error increases.
- The population parameter gets larger.
- The confidence interval gets wider.
6. What argument of the scipy.stats.norm.interval() function can be used to choose the confidence level?
- Alpha (CORRECT)
- scale
- std
- loc
7. Fill in the blank: Because there is more uncertainty involved in estimating the standard error, data professionals use the _____ when working with a small sample size.
- s-distribution
- normal distribution
- t-distribution (CORRECT)
- z-distribution
8. At what sample size does the t-distribution become practically the same as the normal distribution?
- 10
- 5
- 1
- 30 (CORRECT)
9. What would a data professional use to estimate a population parameter using a range of values?
- Interval estimate (CORRECT)
- Point estimate
- Z-score
- Sampling frame
10. What concept describes the likelihood that a particular sampling method will produce a confidence interval that includes the population parameter?
- Confidence level (CORRECT)
- Margin of error
- Sample statistic
- Point estimate
11. A data professional working for a media company is estimating the average amount of time a visitor spends on their website. Based on a sample mean of 4 minutes, they construct the following 95% confidence interval: [3.8 , 4.2]. What does 95% refer to?
- The margin of error
- The percentage of all possible sample means that fall within the range of the interval
- The percentage of data values in the dataset
- The success rate of the estimation process (CORRECT)
12. According to the four steps that detail how to construct a confidence interval for a proportion, which of the following activities are involved in this process? Select all that apply.
- Plot a histogram
- Choose a confidence level (CORRECT)
- Find the margin of error (CORRECT)
- Calculate the interval (CORRECT)
13. A data professional is using scipy.stats.norm.interval() in Python to construct a confidence interval. Which of the following pieces of code can they use to choose a confidence level of 99%?
- scale = 0.99
- std = 0.99
- alpha = 0.99 (CORRECT)
- loc = 0.99
14. A data professional working for a theme park is estimating the mean time visitors spend in the park. They construct the following 95% confidence interval based on a sample mean of 3.5 hours: [2.5, 4.5]. What is the margin of error?
- +/- 4.5 hours
- +/- 1 hour (CORRECT)
- +/- 2.5 hours
- +/- 2 hours
15. Which of the following statements accurately describe the graph of the t-distribution? Select all that apply.
- It has smaller tails than the standard normal distribution.
- As the sample size decreases, the t-distribution approaches the normal distribution.
- It has larger tails than the standard normal distribution. (CORRECT) As the sample size increases, the t-distribution approaches the normal distribution. (CORRECT)
16. Which of the following statements accurately describe a point estimate? Select all that apply.
- A point estimate estimates a sample statistic.
- A point estimate uses a range of values.
- A point estimate estimates a population parameter. (CORRECT)
- A point estimate uses a single value. (CORRECT)
17. In the context of constructing a confidence interval of a population mean, what does the loc argument of the scipy.stats.norm.interval() function refer to?
- Sample standard error
- Sample mean (CORRECT)
- Interquartile range
- Confidence level
18. What shape is the graph of the t-distribution?
- Rectangular shape
- Circular shape
- Square shape
- Bell shape (CORRECT)
19. A data analytics team at a book publisher researches the most popular book subject matter based on sample data. They construct a 95% confidence interval using a sample size of 250. They also construct a 95% confidence interval using a sample size of 5,000. What happens as the sample size increases?
- The confidence interval gets wider.
- The population parameter gets larger.
- The margin of error decreases. (CORRECT)
- The margin of error increases.
20. A data professional at an electricity utility works on a project involving household demand based on sample data. They want to construct a 95% confidence interval using a sample size of 5,000. However, they are unable to get enough data. So they decide to construct a 95% confidence interval using a sample size of 500. What happens as a result of this smaller sample size?
- The margin of error will decrease.
- The population parameter will get larger.
- The confidence interval will get narrower.
- The margin of error will increase. (CORRECT)
21. Fill in the blank: Data professionals use the _____ when working with a small sample size and data that is approximately normally distributed.
- s-distribution
- normal distribution
- t-distribution (CORRECT)
- z-distribution
22. A data professional working for a restaurant chain is constructing a confidence interval to help estimate annual sales. To start, they identify the sample statistic they are working with. According to the four steps that detail how to construct a confidence interval for a proportion, what should they do next?
- Choose a confidence level (CORRECT)
- Calculate the interval
- Plot a histogram
- Find the margin of error
23. Fill in the blank: For small sample sizes, data professionals use the _____ to make calculations with the data.
- normal distribution
- t-distribution (CORRECT)
- z-distribution
- s-distribution
24. What are the main components of a confidence interval? Select all that apply.
- Population parameter
- Confidence level (CORRECT)
- Margin of error (CORRECT)
- Sample statistic (CORRECT)
Correct: A confidence interval is formed by three primary parts: the sample statistic, margin of error, and level of confidence. Confidence intervals are used to indicate the fact that the estimate made in sample data is uncertain.
25. There are four steps involved with constructing a confidence interval. What is typically the first one?
- Identify a sample statistic (CORRECT)
- Choose a confidence level
- Find the margin of error
- Calculate the interval
Correct: Process of constructing confidence interval begins, typically, with sample statistic identification. Then, a confidence level is chosen. The margin of error is determined. Ultimately, the interval is computed.
CONCLUSION – Confidence Intervals
In sum, it’s an exhaustive study of confidence intervals, which is intended to give participants the theoretical background and practical skills required to maneuver through the maze of statistical analysis. By constructing, interpreting, and considering possible complications of confidence intervals, students understand better how to communicate uncertainty in data estimates.
This section presents real-world examples and hands-on applications to prepare participants to understand theoretical principles and apply confidence intervals correctly in practice. Such a statistical tool arms learners for meaningful decisions in many situations involving data analysis and increases their overall skill set in statistical reasoning and data interpretation.