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Official Statistics - Part 3
How to Collect to Publish Data & Its Whole Process as per based RO & RA/SA Syllabus of Prelium Exam 202425
👍,📲,📩,⏭
https://youtu.be/jwrJhnJGIHs
Official Statistics - National & State Statistical System - Gujarati
Statistics India – Social Media Link
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Official Statistics - National & State Statistical System - Gujarati
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In recognition of Prof. C. R. Rao's exceptional contributions to the field of Statistics, the Government of India has established the Prof. C. R. Rao National Award in Statistics to honour young statisticians who have made significant advancements in the field and its applications. Nominations for the 2024-25 award are invited online via the National Awards Portal at https://www.awards.gov.in
The deadline for submitting applications/nominations is February 28.
#MoSPI #NSO #DataForDevelopment #CRRaoNationalAward #YoungStatisticians #StatisticsAward #NationalAwards2024 #StatisticsExcellence #StatisticalContributions #AwardNomination #IndianStatisticians #CRRaoRecognition
Rao Inderjit Singh
NITI Aayog
Press Information Bureau - PIB, Government of India
UGC
Ministry of Education
CBSE
National Council of Educational Research and Training - NCERT
The deadline for submitting applications/nominations is February 28.
#MoSPI #NSO #DataForDevelopment #CRRaoNationalAward #YoungStatisticians #StatisticsAward #NationalAwards2024 #StatisticsExcellence #StatisticalContributions #AwardNomination #IndianStatisticians #CRRaoRecognition
Rao Inderjit Singh
NITI Aayog
Press Information Bureau - PIB, Government of India
UGC
Ministry of Education
CBSE
National Council of Educational Research and Training - NCERT
https://youtu.be/jwrJhnJGIHs
National & State Statistical System Brief video - All Details
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National & State Statistical System Brief video - All Details
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📊 Interval Estimation & Hypothesis Testing 🎯
Before diving into confidence intervals and hypothesis testing, it's essential to understand some statistical fundamentals. Key concepts like probability, probability distributions, Type I & Type II errors, level of significance, statistical power, and confidence intervals form the backbone of statistical analysis.
These principles help in making informed decisions based on data, whether in research, business, or everyday problem-solving. For a quick refresher, check out Appendix A, which covers the essentials!
Confidence Interval (CI): A range of values, derived from sample data, that is likely to contain the true population parameter with a certain probability (e.g., 95% confidence).
Hypothesis Testing: A statistical method used to determine whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis.
Statistical Prerequisites
Before diving into confidence intervals and hypothesis testing, it's essential to understand some statistical fundamentals. Key concepts like probability, probability distributions, Type I & Type II errors, level of significance, statistical power, and confidence intervals form the backbone of statistical analysis.
These principles help in making informed decisions based on data, whether in research, business, or everyday problem-solving. For a quick refresher, check out Appendix A, which covers the essentials!
🔍 Stay tuned for more insights on statistical testing! 📈 #economicscom02#Statistics #HypothesisTesting #ConfidenceIntervals #DataDriven
Before diving into confidence intervals and hypothesis testing, it's essential to understand some statistical fundamentals. Key concepts like probability, probability distributions, Type I & Type II errors, level of significance, statistical power, and confidence intervals form the backbone of statistical analysis.
These principles help in making informed decisions based on data, whether in research, business, or everyday problem-solving. For a quick refresher, check out Appendix A, which covers the essentials!
Confidence Interval (CI): A range of values, derived from sample data, that is likely to contain the true population parameter with a certain probability (e.g., 95% confidence).
Hypothesis Testing: A statistical method used to determine whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis.
Statistical Prerequisites
Before diving into confidence intervals and hypothesis testing, it's essential to understand some statistical fundamentals. Key concepts like probability, probability distributions, Type I & Type II errors, level of significance, statistical power, and confidence intervals form the backbone of statistical analysis.
These principles help in making informed decisions based on data, whether in research, business, or everyday problem-solving. For a quick refresher, check out Appendix A, which covers the essentials!
🔍 Stay tuned for more insights on statistical testing! 📈 #economicscom02#Statistics #HypothesisTesting #ConfidenceIntervals #DataDriven
📊 Hypothesis Testing: The Confidence-Interval Approach
Hypothesis testing using the confidence interval approach helps us determine whether an observed sample estimate is consistent with a proposed hypothesis. If the null hypothesis value falls inside the confidence interval, we do not reject it; if it falls outside, we reject it.
This approach is widely used in econometrics, especially in regression analysis.
1️⃣ Two-Sided (Two-Tail) Test
To illustrate the confidence interval method, let's revisit the wages-education regression model from Equation (3.6.1), where the estimated slope coefficient is:
β^2=0.7240
Suppose we test the hypothesis:
H0:β2=0.5
• H0 assumes that the true effect of education on wages is 0.5.
• H1 suggests that the effect is either lower or higher than 0.5.
Since the alternative hypothesis does not specify a direction, this is a two-sided (two-tail) test.
2️⃣ Decision Rule Using Confidence Intervals
From Equation (5.3.9), we know that the 95% confidence interval for β2\beta_2β2 is:
0.5700 ≤ β2 ≤ 0.8780
🔹 How do we use this interval to test H0H_0H0?
• If β2=0.5 falls within the interval, we do not reject H0.
• If β2=0.5 falls outside the interval, we reject H0.
Since 0.50.50.5 is not inside the interval, we reject H0H_0H0 at the 5% significance level.
3️⃣ Interpretation of Statistical Significance
📌 What does this mean?
• If H0 were true, the probability of observing a value like β^2=0.7240 by pure chance is very low (less than 5%).
• This low probability suggests that our observed β2 is too far from 0.5, leading us to reject H0.
📌 When do we say a result is “statistically significant”?
• If we reject H0, we say the result is statistically significant.
• If we fail to reject H0, the result is not statistically significant.
🔹 Highly Statistically Significant?
• Some researchers say a result is "highly significant" when the Type I error rate (α) is very small (e.g., 1% instead of 5%).
📌 Key Caution:
• Even if we reject H0, there is still a 5% chance (α = 0.05) that we are making a Type I error (rejecting H0 when it is actually true).
🔹 Visualizing the Decision Rule (Figure 5.2):
• Values of β2 inside the confidence interval are plausible under H0.
• Values of β2 outside the confidence interval lead to rejecting H0.
4️⃣ One-Sided (One-Tail) Test
Sometimes, based on economic theory or previous studies, we have a strong expectation that the parameter moves in only one direction.
For example, in the wages-education model, we may hypothesize:
H0:β2≤0.5
Key Differences Between One-Tailed and Two-Tailed Tests:
Feature Two-Tailed Test One-Tailed Test
Hypothesis Form H1:β2≠0.5 H1:β2>0.5
Critical Region Both ends of the distribution Only one end (upper or lower)
Decision Rule Reject H0,if β2 is too high or too low
Reject H0 only if β2 is large enough
🔹 When should we use a one-sided test?
• If prior research strongly suggests that the coefficient can only move in one direction.
• Example: If economic theory suggests more education always increases wages, then we should use a one-tailed test (H1:β2>0.5).
🔹 When should we use a two-sided test?
• If we are unsure about the direction of the effect.
• Example: If we don’t know whether education could increase or decrease wages, we use H1:β2≠0.5.
5️⃣ Key Takeaways
✅ Confidence intervals can be used to test hypotheses
• If the hypothesized value H0 is inside the interval → Do not reject H0.
• If H0 is outside the interval → Reject H0.
✅ A result is “statistically significant” if we reject H0H_0H0.
✅ Two-sided vs. One-sided tests:
• Two-tailed test: Checks if β2 is too high or too low.
• One-tailed test: Checks if β2 is only higher (or lower) than expected.
✅ Even if we reject H0, there is always a small chance (Type I error) that we are wrong!
✅ Final Summary
✔️ Two-tailed tests check for deviations in both directions (higher or lower).
✔️ One-tailed tests check for deviations in one specific direction.
✔️ Confidence intervals provide a simple decision rule for hypothesis testing.
Hypothesis testing using the confidence interval approach helps us determine whether an observed sample estimate is consistent with a proposed hypothesis. If the null hypothesis value falls inside the confidence interval, we do not reject it; if it falls outside, we reject it.
This approach is widely used in econometrics, especially in regression analysis.
1️⃣ Two-Sided (Two-Tail) Test
To illustrate the confidence interval method, let's revisit the wages-education regression model from Equation (3.6.1), where the estimated slope coefficient is:
β^2=0.7240
Suppose we test the hypothesis:
H0:β2=0.5
• H0 assumes that the true effect of education on wages is 0.5.
• H1 suggests that the effect is either lower or higher than 0.5.
Since the alternative hypothesis does not specify a direction, this is a two-sided (two-tail) test.
2️⃣ Decision Rule Using Confidence Intervals
From Equation (5.3.9), we know that the 95% confidence interval for β2\beta_2β2 is:
0.5700 ≤ β2 ≤ 0.8780
🔹 How do we use this interval to test H0H_0H0?
• If β2=0.5 falls within the interval, we do not reject H0.
• If β2=0.5 falls outside the interval, we reject H0.
Since 0.50.50.5 is not inside the interval, we reject H0H_0H0 at the 5% significance level.
3️⃣ Interpretation of Statistical Significance
📌 What does this mean?
• If H0 were true, the probability of observing a value like β^2=0.7240 by pure chance is very low (less than 5%).
• This low probability suggests that our observed β2 is too far from 0.5, leading us to reject H0.
📌 When do we say a result is “statistically significant”?
• If we reject H0, we say the result is statistically significant.
• If we fail to reject H0, the result is not statistically significant.
🔹 Highly Statistically Significant?
• Some researchers say a result is "highly significant" when the Type I error rate (α) is very small (e.g., 1% instead of 5%).
📌 Key Caution:
• Even if we reject H0, there is still a 5% chance (α = 0.05) that we are making a Type I error (rejecting H0 when it is actually true).
🔹 Visualizing the Decision Rule (Figure 5.2):
• Values of β2 inside the confidence interval are plausible under H0.
• Values of β2 outside the confidence interval lead to rejecting H0.
4️⃣ One-Sided (One-Tail) Test
Sometimes, based on economic theory or previous studies, we have a strong expectation that the parameter moves in only one direction.
For example, in the wages-education model, we may hypothesize:
H0:β2≤0.5
Key Differences Between One-Tailed and Two-Tailed Tests:
Feature Two-Tailed Test One-Tailed Test
Hypothesis Form H1:β2≠0.5 H1:β2>0.5
Critical Region Both ends of the distribution Only one end (upper or lower)
Decision Rule Reject H0,if β2 is too high or too low
Reject H0 only if β2 is large enough
🔹 When should we use a one-sided test?
• If prior research strongly suggests that the coefficient can only move in one direction.
• Example: If economic theory suggests more education always increases wages, then we should use a one-tailed test (H1:β2>0.5).
🔹 When should we use a two-sided test?
• If we are unsure about the direction of the effect.
• Example: If we don’t know whether education could increase or decrease wages, we use H1:β2≠0.5.
5️⃣ Key Takeaways
✅ Confidence intervals can be used to test hypotheses
• If the hypothesized value H0 is inside the interval → Do not reject H0.
• If H0 is outside the interval → Reject H0.
✅ A result is “statistically significant” if we reject H0H_0H0.
✅ Two-sided vs. One-sided tests:
• Two-tailed test: Checks if β2 is too high or too low.
• One-tailed test: Checks if β2 is only higher (or lower) than expected.
✅ Even if we reject H0, there is always a small chance (Type I error) that we are wrong!
✅ Final Summary
✔️ Two-tailed tests check for deviations in both directions (higher or lower).
✔️ One-tailed tests check for deviations in one specific direction.
✔️ Confidence intervals provide a simple decision rule for hypothesis testing.
✔️ Statistical significance does not mean absolute certainty—it means the result is unlikely due to random chance.
✔️ Use prior research and theory to determine whether to use a one-tailed or two-tailed test.
#economicscom02#Statistics #HypothesisTesting #ConfidenceIntervals #DataScience #Econometrics #MachineLearning #DataAnalysis #RegressionModeling
✔️ Use prior research and theory to determine whether to use a one-tailed or two-tailed test.
#economicscom02#Statistics #HypothesisTesting #ConfidenceIntervals #DataScience #Econometrics #MachineLearning #DataAnalysis #RegressionModeling
