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Questions and Answers
What formula correctly represents the variance of the sample mean, denoted as $σ_\bar{y}$?
What formula correctly represents the variance of the sample mean, denoted as $σ_\bar{y}$?
- $σ_y / \sqrt{n}$
- $σ_y^2 / n$ (correct)
- $σ_y^2 / \sqrt{n}$
- $σ_y$
What condition must an estimator $\hat{μ}_y$ of the population value $μ_y$ satisfy to be considered consistent?
What condition must an estimator $\hat{μ}_y$ of the population value $μ_y$ satisfy to be considered consistent?
- $\bar{Y} \xrightarrow{p} 0$
- $P(\hat{μ}_y \rightarrow μ_y)$ (correct)
- $\bar{Y}$ is normally distributed.
- Its root mean square error is the smallest possible.
What does a large p-value suggest in the context of hypothesis testing?
What does a large p-value suggest in the context of hypothesis testing?
- The observed value $\bar{Y}$ is consistent with the null hypothesis. (correct)
- A large t-statistic.
- A large $\bar{Y}$.
- Rejection of the null hypothesis.
Why is the Central Limit Theorem (CLT) considered a fundamental concept in statistics?
Why is the Central Limit Theorem (CLT) considered a fundamental concept in statistics?
If a sample of size 81 is drawn from a population with a standard deviation of 81, what is the standard error of the mean?
If a sample of size 81 is drawn from a population with a standard deviation of 81, what is the standard error of the mean?
A random sample of 15 is taken from a normally distributed population with a sample mean of 75 and known sample variance of 25. What is the upper limit of a 95% confidence interval for the population mean?
A random sample of 15 is taken from a normally distributed population with a sample mean of 75 and known sample variance of 25. What is the upper limit of a 95% confidence interval for the population mean?
When multiple unbiased estimators are available for estimating a population mean, which criterion should guide the selection of the most appropriate estimator?
When multiple unbiased estimators are available for estimating a population mean, which criterion should guide the selection of the most appropriate estimator?
In Brussels, data from 400 school districts yields an average fifth-grade test score of 712.1 with a standard deviation of 23.2. Construct a 95% confidence interval for the mean test score in the population.
In Brussels, data from 400 school districts yields an average fifth-grade test score of 712.1 with a standard deviation of 23.2. Construct a 95% confidence interval for the mean test score in the population.
Flashcards
What is σ²Ῠ?
What is σ²Ῠ?
Variance of the sample mean. Denoted as σ²Ῠ.
What is a consistent estimator?
What is a consistent estimator?
An estimator that gets closer to the true population value as the sample size increases.
What does a large p-value imply?
What does a large p-value imply?
Indicates that the observed data is likely under the null hypothesis.
Why is the Central Limit Theorem important?
Why is the Central Limit Theorem important?
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What is the standard error of the mean?
What is the standard error of the mean?
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Estimator choosing
Estimator choosing
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Confidence interval
Confidence interval
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What is a Small Class Size?
What is a Small Class Size?
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Study Notes
- Variance of 𝑌̅, 𝜎𝑌̅, is given by the formula 𝜎𝑌2 /𝑛.
- An estimator 𝜇𝑌̂ of the population value 𝜇𝑌 is consistent if 𝜇𝑌̂ → 𝜇𝑌.
- A large p-value implies that the observed value 𝑌̅ is consistent with the null hypothesis.
- The central limit theorem is important because, for any population, it indicates the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population.
- If a sample of size 81 is taken from a population whose standard deviation is equal to 81, the standard error of the mean is equal to 9.
- From a random sample of size 15 with a sample mean of 75 and a sample variance of 25, the upper limit of a 95% confidence interval for the population mean is 77.53.
- When estimating a population mean, if there is more than one unbiased estimator, the criteria for choosing between the estimators is the one with the smallest population variance.
Practice Question 1
- Data on fifth-grade test scores for 400 school districts in Brussels revealed an average score 𝑌̅ of 712.1 and a standard deviation 𝑠𝑌 of 23.2.
- For districts divided into small classes (< 20 students per teacher) and large classes (≥ 20 students per teacher):
- Small classes: Average score = 721.8, Standard Deviation = 24.4, n = 150.
- Large classes: Average Score = 710.9, Standard Deviation = 20.6, n = 250.
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