COURSE 4: THE POWER OF STATISTICS

Module 4: Confidence Intervals

GOOGLE ADVANCED DATA ANALYTICS PROFESSIONAL CERTIFICATE

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TABLE OF CONTENT

INTRODUCTION – Confidence Intervals

Throughout this section, participants will delve into the crucial concept of confidence intervals, gaining a comprehensive understanding of how data professionals employ them to convey the uncertainty inherent in their estimates. The module not only equips learners with the practical skills to construct and interpret confidence intervals but also emphasizes the importance of avoiding common misinterpretations that can arise in the process.

By providing real-world examples and practical applications, this overview ensures that participants not only grasp the theoretical underpinnings of confidence intervals but also develop the proficiency to apply this statistical tool accurately in diverse data analysis scenarios. This comprehensive exploration serves as an invaluable resource for those seeking to enhance their statistical literacy and make informed decisions based on the uncertainty inherent in data estimates.

Learning Objectives

  • Use Python to construct a confidence interval
  • Describe how to construct a confidence interval for means and proportions
  • Identify common forms of misinterpretation associated with confidence intervals
  • Describe how to properly interpret a confidence interval
  • Define concepts related to confidence intervals such as confidence level and margin of error
  • Explain the difference between a point estimate and an interval estimate

PRACTICE QUIZ: TEST YOUR KNOWLEDGE: INTRODUCTION TO CONFIDENCE INTERVALS

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: An interval estimate uses a range of values to estimate a population parameter.

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: Margin of error is the maximum expected difference between a population parameter and a sample estimate.

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 refers to the success rate of the estimation process, not that the values in the dataset fall within the interval. This is a common misconception.

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: When constructing a confidence interval, first, identify a sample statistic; second, choose a confidence level; third, find the margin of error; and fourth, calculate 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: The interval lies between 46 seconds and 54 seconds. The lower limit of the interval is the sample mean minus the margin of error: 50 – 4 = 46. The upper limit of the interval is the sample mean plus the margin of error: 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: As the sample size gets larger, the confidence interval narrows. This is because as the sample size increases, the margin of error decreases. If every member of the population could be sampled, the margin of error would be zero.

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: The pandas.DataFrame.std() function, which returns the standard deviation, enables a data professional to compute the standard deviation term in the sample standard error of a mean. Sample standard error is the sample standard deviation divided 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: They should specify confidence (alpha), which they set to the confidence level; loc, which they set to the sample mean; and scale, which they set to 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.

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?

  • 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: The main components of a confidence interval are a sample statistic, margin of error, and confidence level. Confidence intervals help express the uncertainty of an estimate based on sample data.

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: Feedback: Typically, the first step of constructing a confidence interval is identifying a sample statistic. Next, a confidence level is chosen. Then, the margin of error is found. Finally, the interval is calculated.

CONCLUSION – Confidence Intervals

In conclusion, this comprehensive exploration of confidence intervals equips participants with the essential knowledge and practical skills needed to navigate the intricacies of statistical analysis. By delving into the construction, interpretation, and potential pitfalls of confidence intervals, learners gain a robust understanding of how to effectively convey the uncertainty associated with data estimates.

The real-world examples and hands-on applications provided in this section ensure that participants not only comprehend the theoretical foundations but also develop the proficiency to apply confidence intervals with precision. Armed with this statistical tool, learners are well-prepared to make informed decisions in various data analysis contexts, contributing to their overall proficiency in statistical reasoning and data interpretation.

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