Estimating the Mean

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The mean is the first statistic we learn, the cornerstone of many analyses. But the question is how well do we understand its estimation? For statisticians, estimating the mean is more than just summing and dividing. It involves navigating assumptions, choosing appropriate methods, and understanding the implications of our choices. Let us delve deeper into the art and science of estimating the mean.

The Simple Sample Mean: A Foundation

The Formula of sample mean $\overline{x}= \frac{\sum\limits_{i=1}^n x_i}{n}$​​. The sample mean is the unbiased estimator of population mean ($\mu$) under ideal conditions (simple random sampling, independent and identically distributed data). Violating the assumption can lead to biased estimates. For large samples, the distribution of the sample mean approximates a normal distribution, regardless of the population distribution due to the Central Limit Theorem (CLT).

Weighted Means

Beyond Simple Random Sampling, For weighted means, observations have varying importance (e.g., survey data with different sampling weights). The formula of weighted mean is $ \overline{x}_w = \frac{\sum\limits_{i=1}^n w_ix_i}{\sum\limits_{i=1}^n w_i}$. Weighted means are used in Survey sampling, and dealing with non-response. In Stratified Sampling, estimate the mean when the population is divided into strata for getting reduced variance, and improved precision. In cluster sampling have unique challenges of estimating the mean with cluster sampling, where observations are grouped.

Robust Estimation

Robust Estimation is required when the sample mean is vulnerable to extreme values. The alternative of the sample mean is the median which emphasizes its robustness to outliers. The trimmed mean is also used to balance out the robustness and efficiency.

Confidence Intervals for Estimating the Mean

Confidence Intervals make use of standard error to estimate the mean to reflect the precision of the estimate. For small samples, t-distribution while for large samples, z-distribution is used for the construction of confidence intervals. Bootstrapping (a non-parametric method) can also be used for constructing confidence intervals, especially, useful when assumptions are violated.

Point Estimate: To estimate the population mean $\mu$ for a random variable $x$ using a sample of values, the best possible point estimate is the sample mean $\overline{x}$.

Interval Estimate: An interval estimate for mean $\mu$ is constructed by starting with sample mean $\overline{x}$ and adding a margin of error (S.E.) above and below the mean $\overline{x}$. The interval is of the form $(\overline{x} – SE, \overline{x} + SE)$.

Example: Suppose that the mean height of Pakistani men is between 67.5 and 70.5 inches with a level of confidence of $c = 0.90$. To estimate the men’s height, the sample mean $\overline{x}$ is 69 inches with a margin of error = 1.5 inches. That is, $(\overline{x} – SE, \overline{x}+SE) = (69 – 1.5, 69+1.5) = (67.5, 70.5)$.

Note that the margin of error used for constructing an interval estimate depends on the level of confidence interval. A larger level of confidence will result in a larger margin of error and hence a wider interval.

Estimating the Mean boxplot with mean

Calculating Margin of Error for a Large Sample Data

If a random variable $x$ is normally distributed (with a known population standard deviation $\sigma$) or if the sample size $n$ is at least 30 (we will apply Central Limit Theorem, which will guarantee that),

  • $\overline{x}$ is approximately normally distributed
  • $\mu_{\overline{x}} = \mu$
  • $\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}}$

The mean value of $\overline{x}$ equals the estimated population mean $\mu$. Given the desired level of confidence $c$, it is try to find the amount of error $E$ necessary to ensure that the probability of $\overline{x}$ being within $E$ of the mean is $c$.

There are always two critical $Z$-scores ($\pm z_c$ which give the appropriate probability for the standard normal distribution), and the corresponding probability for the distribution of $\overline{x}$ is $z_c \times \sigma_{\overline{x}}$ or

$$E=z_c \frac{\sigma}{\sqrt{n}}$$

Usually, $\sigma$ is unknown, but if $n\ge 30$ then the sample standard deviation $s$ is generally a reasonable estimate.

Estimating the Mean Histogram

Dealing with Missing Data

When dealing with missing data, one can impute mean. Imputing the mean is simple but it can underestimate variance. One can also perform multiple imputations to account for the uncertainty.

Bayesian Estimation

In Bayesian estimation, the prior and posterior distributions are used for estimating the mean by incorporating prior information, updated beliefs about the mean, and handling uncertainty.

Summary

Estimating the mean is a fundamental statistical task, but it requires careful consideration of assumptions, data characteristics, and the goals of the analysis. By understanding the nuances of different estimation methods, statisticians can provide more accurate and reliable insights.

Exploratory Data Analysis in R Language

Evaluating Regression Models Quiz 11

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The post is about Evaluating Regression Models Quiz with answers. There are 20 multiple-choice questions about regression models and their evaluation, covering regression analysis, assumptions of regression, coefficient of determination, predicted and predictor variables, etc. Let us start with the Evaluating Regression Models Quiz now.

Evaluating Regression Models Quiz

Online MCQs about Evaluating Regression Models

1. The test used to test the individual partial coefficient in the multiple regression is

 
 
 
 

2. Which situations are helped by using the cross-validation method to train your model?

 
 
 
 

3. A third-order polynomial regression model is described as which of the following?

 
 
 
 

4. One cannot apply test of significance if $\varepsilon_i$ in the model $y_i = \alpha + \beta X_i+\varepsilon_i$ are

 
 
 
 

5. An underfit model is said to have which of the following?

 
 
 
 

6. The ratio of explained variation to the total variation of the following regression model is called $y_i = \beta_0 + \beta_1 x_{1i} + \beta_2x_{2i} + \varepsilon_i, \quad i=1,2,\cdots, n$.

 
 
 
 

7. Regression coefficients may have the wrong sign for the following reasons

 
 
 
 

8. What does regularization introduce into a model that results in a drop in variance?

 
 
 
 

9. A training set is ————–.

 
 
 
 

10. When using the poly() function to fit a polynomial regression model, you must specify “raw = FALSE” so you can get the expected coefficients.

 
 

11. What is the difference between Ridge and Lasso regression?

 
 
 
 

12. When evaluating models, what is the term used to describe a situation where a model fits the training data very well but performs poorly when predicting new data?

 
 
 
 

13. How can the following plot be used to see if residuals satisfy the requirements for a linear regression?

Evaluating Regression Models Quiz 11

 
 
 
 

14. When tuning a model, a grid search attempts to find the value of a parameter that has the smallest —————-.

 
 
 
 

15. When we fit a linear regression model we make strong assumptions about the relationships between variables and variance. These assumptions need to be assessed to be valid if we are to be confident in estimated model parameters. The questions below will help ascertain that you know what assumptions are made and how to verify these.

Which of these is not assumed when fitting a linear regression model?

 
 
 
 

16. Parveen previously fitted a linear regression model to quantify the relationship between age and lung function measured by FEV1. After she fitted her linear regression model she decided to assess the validity of the linear regression assumptions. She knew she could do this by assessing the residuals and so produced the following plot known as a QQ plot.

QQ Plot Regression model residuals

How can she use this plot to see if her residuals satisfy the requirements for a linear regression?

 
 
 
 

17. The residuals are the distance between the observed values and the fitted regression line. If the assumptions of linear regression hold how would we expect the residuals to behave?

 
 
 
 

18. What is a strategy you can employ to address an underfit model?

 
 
 
 

19. Let the value of the $R^2$ for a model is 0.0104. What does this tell?

 
 
 

20. A testing set is —————.

 
 
 
 

MCQs Evaluating Regression Models Quiz with Answers

  • When using the poly() function to fit a polynomial regression model, you must specify “raw = FALSE” so you can get the expected coefficients.
  • A third-order polynomial regression model is described as which of the following?
  • When evaluating models, what is the term used to describe a situation where a model fits the training data very well but performs poorly when predicting new data?
  • An underfit model is said to have which of the following?
  • What does regularization introduce into a model that results in a drop in variance?
  • When tuning a model, a grid search attempts to find the value of a parameter that has the smallest —————-.
  • Which situations are helped by using the cross-validation method to train your model?
  • What is a strategy you can employ to address an underfit model?
  • What is the difference between Ridge and Lasso regression?
  • A training set is ————–.
  • A testing set is —————.
  • Regression coefficients may have the wrong sign for the following reasons
  • The ratio of explained variation to the total variation of the following regression model is called $y_i = \beta_0 + \beta_1 x_{1i} + \beta_2x_{2i} + \varepsilon_i, \quad i=1,2,\cdots, n$.
  • One cannot apply test of significance if $\varepsilon_i$ in the model $y_i = \alpha + \beta X_i+\varepsilon_i$ are
  • The test used to test the individual partial coefficient in the multiple regression is
  • When we fit a linear regression model we make strong assumptions about the relationships between variables and variance. These assumptions need to be assessed to be valid if we are to be confident in estimated model parameters. The questions below will help ascertain that you know what assumptions are made and how to verify these. Which of these is not assumed when fitting a linear regression model?
  • Parveen previously fitted a linear regression model to quantify the relationship between age and lung function measured by FEV1. After she fitted her linear regression model she decided to assess the validity of the linear regression assumptions. She knew she could do this by assessing the residuals and so produced the following plot known as a QQ plot. How can she use this plot to see if her residuals satisfy the requirements for a linear regression?
  • How can the following plot be used to see if residuals satisfy the requirements for a linear regression?
  • Let the value of the $R^2$ for a model is 0.0104. What does this tell?
  • The residuals are the distance between the observed values and the fitted regression line. If the assumptions of linear regression hold how would we expect the residuals to behave?
Evaluating Regression Models Quiz

Performing Statistical Models in R

MCQs Big Data Questions 2

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The post is about MCQs Big Data Questions with Answers. There are 20 multiple-choice questions with answers. “Ready to test your big data knowledge? Take a quiz today and see how you fare! Share your results in the comments and let us know what topics you’d like to see covered in future quizzes.” Let us start with the Online MCQs Big Data Questions now.

Please go to MCQs Big Data Questions 2 to view the test

MCQs Big Data Questions

Online MCQs Big Data Questions with Answers
  • Which is the most compelling reason why mobile advertising is related to big data?
  • Which of the following summarizes the process of using data streams?
  • These two characteristics define the ratio between populated and unpopulated cells in a data source.
  • What does it mean for a device to be “smart”?
  • At-rest and in-transit data each have unique security concerns.
  • What does the term “in situ” mean in the context of big data?
  • What are data silos and why are they bad?
  • —————— is a measure of how fast the data is coming in.
  • These two characteristics are critical to implementing a successful high-velocity data strategy
  • What are the steps required for data analysis?
  • Which of the following is a technique mentioned in the videos for building a model?
  • Which of the Big Data processing tools provides distributed storage and processing of Big Data?
  • What does the attribute “Veracity” imply in the context of Big Data?
  • Defining the ————– ————– is the first step in any big data strategy.
  • A well-defined and comprehensive big data strategy makes the benefits of big data ————— for the organization.
  • What are the ways to address data quality issues?
  • Data in a data lake is most commonly stored in its natural or raw form.
  • What is the benefit of using pre-built Hadoop images?
  • Which of the following are general requirements for a programming language to support big data models?
  • Which of the following is the best description of why it is important to learn about the foundations of big data?

R Programming Language