Podcast
Questions and Answers
The difference between an estimator's expected value and the true value of the parameter being estimated is called ______.
The difference between an estimator's expected value and the true value of the parameter being estimated is called ______.
bias
Unbiased estimators guarantee that the sample mean ($\overline{x}$) will always be exactly equal to the population mean ($μ$).
Unbiased estimators guarantee that the sample mean ($\overline{x}$) will always be exactly equal to the population mean ($μ$).
False (B)
Which of the following statements best describes the relationship between confidence level and interval width?
Which of the following statements best describes the relationship between confidence level and interval width?
- Higher confidence levels lead to narrower intervals.
- Lower confidence levels lead to wider intervals.
- Higher confidence levels lead to wider intervals. (correct)
- Confidence level and interval width are unrelated.
State the Central Limit Theorem in your own words.
State the Central Limit Theorem in your own words.
When calculating the required sample size for estimating a population mean, which of the following factors does NOT affect the result?
When calculating the required sample size for estimating a population mean, which of the following factors does NOT affect the result?
What happens to the width of a confidence interval as the sample size increases, assuming all other factors remain constant?
What happens to the width of a confidence interval as the sample size increases, assuming all other factors remain constant?
When is it appropriate to use the t-distribution instead of the z-distribution when constructing a confidence interval for a population mean?
When is it appropriate to use the t-distribution instead of the z-distribution when constructing a confidence interval for a population mean?
In the context of confidence intervals for proportions, what condition must be met to justify using the normal approximation?
In the context of confidence intervals for proportions, what condition must be met to justify using the normal approximation?
Explain the difference between a 'prediction interval' and a 'confidence interval'.
Explain the difference between a 'prediction interval' and a 'confidence interval'.
A ______ is an interval that is likely to contain a specified proportion of the population.
A ______ is an interval that is likely to contain a specified proportion of the population.
Which of the following is the correct way to calculate the confidence interval for the standard deviation ($\sigma$) of a normal population?
Which of the following is the correct way to calculate the confidence interval for the standard deviation ($\sigma$) of a normal population?
In hypothesis testing, the primary goal of researchers is to:
In hypothesis testing, the primary goal of researchers is to:
A one-tailed hypothesis test is used exclusively when the alternative hypothesis states ''.
A one-tailed hypothesis test is used exclusively when the alternative hypothesis states ''.
What does the significance level ($\alpha$) represent in hypothesis testing?
What does the significance level ($\alpha$) represent in hypothesis testing?
Which of the following reduces the probability of committing a Type II error?
Which of the following reduces the probability of committing a Type II error?
The probability of avoiding a Type II error is referred to as ______.
The probability of avoiding a Type II error is referred to as ______.
Which of the following assumptions is essential for conducting a one-sample z-test?
Which of the following assumptions is essential for conducting a one-sample z-test?
Describe the difference between a null and alternative hypotheses.
Describe the difference between a null and alternative hypotheses.
When should a one-sample t-test be used instead of a one-sample z-test?
When should a one-sample t-test be used instead of a one-sample z-test?
Which distribution is approximated when conducting One Sample z-test for Proportions?
Which distribution is approximated when conducting One Sample z-test for Proportions?
What is the assumption when conducting Two Sample z-test?
What is the assumption when conducting Two Sample z-test?
How to test if data has equal variances?
How to test if data has equal variances?
Match is the following:
Match is the following:
What does a point estimate represent in statistical inference?
What does a point estimate represent in statistical inference?
Which of the following describes 'Inferential Statistics'?
Which of the following describes 'Inferential Statistics'?
Estimation accuracy increases with large samples, but there is still no reason we should expect a point estimate from a given sample to be exactly equal to the population parameter it is supposed to estimate.
Estimation accuracy increases with large samples, but there is still no reason we should expect a point estimate from a given sample to be exactly equal to the population parameter it is supposed to estimate.
______ is an interval within which the value of a parameter of a population has a stated probability of occurring.
______ is an interval within which the value of a parameter of a population has a stated probability of occurring.
Ideally, we prefer a _____ interval with a ______ degree of confidence.
Ideally, we prefer a _____ interval with a ______ degree of confidence.
Central Limit Theorem is only used if the population is normally distributed.
Central Limit Theorem is only used if the population is normally distributed.
A ______ is likely to contain the value of an item sampled from a population at a future time.
A ______ is likely to contain the value of an item sampled from a population at a future time.
Which of the following distribution applies to Paired Sample t-test?
Which of the following distribution applies to Paired Sample t-test?
The null hypothesis is the hypothesis that the researcher is trying to prove.
The null hypothesis is the hypothesis that the researcher is trying to prove.
What is another name for Type I error?
What is another name for Type I error?
What is the purpose of the homogeneity of variances test (e.g., Levene's test) in the context of a two-sample t-test?
What is the purpose of the homogeneity of variances test (e.g., Levene's test) in the context of a two-sample t-test?
A paired sample t-test is appropriate when the two samples are independent of each other.
A paired sample t-test is appropriate when the two samples are independent of each other.
Which of the following is the correct formula for computing Z test?
Which of the following is the correct formula for computing Z test?
The claim that the students in his school are above average intelligence is also known as?
The claim that the students in his school are above average intelligence is also known as?
Flashcards
Inferential Statistics
Inferential Statistics
Methods for making generalizations about a population from a sample.
Point Estimate
Point Estimate
A single value estimate representing a population parameter.
Bias (in estimation)
Bias (in estimation)
The difference between an estimator's expected value and the true parameter value.
Unbiased Estimator
Unbiased Estimator
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Interval Estimate
Interval Estimate
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Standard Error
Standard Error
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Margin of Error
Margin of Error
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Central Limit Theorem
Central Limit Theorem
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Prediction Interval
Prediction Interval
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Tolerance Interval
Tolerance Interval
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Hypothesis
Hypothesis
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Null Hypothesis (Ho)
Null Hypothesis (Ho)
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Alternative Hypothesis (Ha)
Alternative Hypothesis (Ha)
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One-Tailed Hypothesis Testing
One-Tailed Hypothesis Testing
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Two-Tailed Hypothesis Testing
Two-Tailed Hypothesis Testing
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Type I Error (False Positive or α)
Type I Error (False Positive or α)
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Type II Error (False Negative or β)
Type II Error (False Negative or β)
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Statistical Power
Statistical Power
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One Sample z-test
One Sample z-test
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One Sample t-test
One Sample t-test
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One Sample z-test for Proportions
One Sample z-test for Proportions
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Two Sample z-test
Two Sample z-test
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Two Sample t-test (Equal Variance)
Two Sample t-test (Equal Variance)
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Two Sample t-test (Unequal Variance)
Two Sample t-test (Unequal Variance)
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Test for Homogeneity of Variances
Test for Homogeneity of Variances
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Paired Sample t-test
Paired Sample t-test
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Two Sample z-test for Proportions
Two Sample z-test for Proportions
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Study Notes
One Sample Estimation
- Inferential statistics involves making inferences or generalizations about a population based on samples.
- A point estimate is a single value estimating a population parameter, like using the sample mean (𝑥ҧ) to estimate the population mean (μ).
- Estimators are not expected to be perfectly accurate.
- Bias refers to the difference between the estimator's expected value and the true parameter value.
- An unbiased estimator has zero bias, examples include μ = xത and σ2 = s2.
- Estimation accuracy typically increases with larger samples.
- A point estimate from a given sample is unlikely to exactly match the population parameter.
Interval Estimate
- An interval estimate is a range within which a population parameter is likely to occur with a stated probability.
- Confidence intervals with σ known are calculated using the formula: xത − z α/2 (σ/√n) < μ < xത + z α/2 (σ/√n).
- Where σ/√n is the Standard Error, and z α/2 (σ/√n) is the Margin of Error
Common Confidence Level and Z scores
- For a 95% confidence interval, the z-score is 1.96.
- For a 99% confidence interval, the z-score is 2.58.
- A shorter interval with a high degree of confidence is preferable.
Confidence Intervals with σ unknown but n ≥ 30
- Confidence intervals are calculated using: xത − z α/2 (s/√n) < μ < xത + z α/2 (s/√n).
- As sample size increases, the distribution of frequencies approximates a normal distribution curve according to the Central Limit Theorem. The central Limit Theorem is applicable regardless of the population distribution.
Central Limit Theorem
- If a large enough sample is drawn from a population, the distribution of the sample mean is approximately normal.
Sample Size Determination
- To estimate μ with 100(1 − α)% confidence, the error should not exceed a specified amount e, the sample size is calculated as: n = (Zα/2 * σ / e)^2.
- Always round fractional sample sizes up to the next whole number to maintain the desired confidence level.
Confidence Intervals with σ unknown but n ≥ 30
- Confidence intervals are calculated using: xത − z α/2 (s/√n) < μ < xത + z α/2 (s/√n).
Confidence Intervals with σ unknown but n < 30
- Confidence intervals are calculated using: xത − t α/2 (s/√n) < μ < xത + t α/2 (s/√n).
- William Sealy Gosset, known as "Student," discovered the t-distribution and its use.
Confidence Intervals for Proportions with Large Sample Size
- Confidence intervals are calculated using: pො − z α/2 √(pොqො/n) < μ < pො + z α/2 √(pොqො/n).
- np and n(1-p) must both be greater than or equal to 10 to justify using the normal approximation.
Prediction Intervals
- A prediction interval is likely to contain the value of an item sampled from a population at a future time.
- Calculated as: xത − t α/2 * s * √(1 + 1/n) < μ < xത + t α/2 * s * √(1 + 1/n).
- This method is sensitive to the assumption that the population is normal.
Tolerance Intervals
- A tolerance interval is likely to contain a specified proportion of the population.
- Calculated as: xത − k(n,α,γ)s < μ < xത + k(n,α,γ)s.
Confidence Intervals for the Variance of a Normal Population
- Confidence intervals for the standard deviation σ are found by taking the square roots of the confidence bounds for the variance.
One Sample Hypothesis Testing
- A hypothesis is an assumption about a population parameter.
- Hypothesis testing involves a null hypothesis (Ho) and an alternative hypothesis (Ha).
- Researchers aim to reject, nullify, or disprove the null hypothesis.
Types of Hypothesis Testing
- One-tailed hypothesis testing involves an alternative hypothesis stated as < or >.
- Two-tailed hypothesis testing is used when the alternative hypothesis is expressed as ≠.
Types of Error
- Type I error (false positive/α) occurs when the null hypothesis is rejected when it is true.
- The probability of committing a Type I error is the significance level.
- Type II error (false negative/β) occurs when the null hypothesis is not rejected when it is false.
- Statistical power is the probability of avoiding a Type II error.
- Increasing sample size, increasing the significance level, and minimizing random errors can reduce the risk of a Type II error.
One Sample z-test
- Assumptions include: continuous data, normal distribution, and known population standard deviation or large sample size.
- Calculated as: 𝑧 = (𝑥ഥ − 𝜇) / (σ/√n) or 𝑧 = (𝑥ഥ − 𝜇) / (s/√n).
One Sample t-test
- Assumptions include: continuous data, normal distribution, and unknown population standard deviation with a small sample size.
- Calculated as: 𝑡 = (𝑥ഥ − 𝜇) / (s/√n).
One Sample z-test for Proportions
- Assumptions include: population follows a binomial distribution, and both mean (np) and variance (n(1-p)) are greater than 10.
- Calculated as: 𝑧 = (𝑝ො − 𝑝) / √(pq/n).
Two Sample Hypothesis Testing
Two Sample z-test
- Assumptions include: continuous data, independent samples, normal distribution, and known population standard deviation or large sample size.
- Calculated as: 𝑍 = (𝑥ҧ1 − 𝑥ҧ2 − (𝜇1 − 𝜇2)) / √(σ1^2/n1 + σ2^2/n2) or 𝑍 = (𝑥ҧ1 − 𝑥ҧ2 − (𝜇1 − 𝜇2)) / √(s1^2/n1 + s2^2/n2).
Two Sample t-test (Equal Variance)
- Assumptions include: continuous data, independent samples, normal distribution, and unknown population standard deviation with a small sample size.
- Calculated as: t = (xത1 − xത2 − (μ1 − μ2)) / (Sp * √(1/n1 + 1/n2)).
Two Sample t-test (Unequal Variance)
- Assumptions include: continuous data, independent samples, normal distribution, and unknown population standard deviation with a small sample size.
- Calculated as: t = (xത1 − xത2 − (μ1 − μ2)) / √(s1^2/n1 + s2^2/n2).
Test for Homogeneity of Variances
- Ho: Data has equal variances, Ha: Data has unequal variances.
- Levene’s Test is used if data is normally distributed, while Bartlett’s Test is used if data is not normally distributed.
- Determines which type of t-test to use.
Paired Sample t-test
- Assumptions include: continuous data, dependent samples, normal distribution, and unknown population standard deviation with a small sample size.
- Calculated as: t = (Dഥ − μD) / (SD/√n).
- Where Dഥ = (σ D) / n and SD = √(σ D^2 − (σ D)^2 / n ) / (n − 1).
Two Sample z-test for Proportions
- Assumptions include: population follows a binomial distribution, and both mean (np) and variance (n(1-p)) are greater than 10.
- Calculated as: Z = (pො1 − pො2 − (p1 − p2)) / √(pq (1/n1 + 1/n2)).
- Where pത = (x1 + x2) / (n1 + n2), qത = 1 − pത, pො1 = x1 / n1, and pො2 = x2 / n2.
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