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Questions and Answers
What is the appropriate condition to use a Welch t-test instead of a pooled t-test?
What is the appropriate condition to use a Welch t-test instead of a pooled t-test?
When should log transformation be considered for datasets?
When should log transformation be considered for datasets?
What does the ideal result of a log transformation include?
What does the ideal result of a log transformation include?
How is the standard error for the samples’ mean difference calculated in a Welch t-test?
How is the standard error for the samples’ mean difference calculated in a Welch t-test?
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What occurs when the log-transformed data are symmetric?
What occurs when the log-transformed data are symmetric?
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What is the purpose of using the t-ratio instead of the z-ratio?
What is the purpose of using the t-ratio instead of the z-ratio?
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In the formula for the standard error of the sample mean, $SE(\bar{Y})$, what does 'n' represent?
In the formula for the standard error of the sample mean, $SE(\bar{Y})$, what does 'n' represent?
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Under the Central Limit Theorem (CLT), which distribution is the sample mean approximately normally distributed?
Under the Central Limit Theorem (CLT), which distribution is the sample mean approximately normally distributed?
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What is the term for the number of independent values used to estimate $SE(\bar{Y})$?
What is the term for the number of independent values used to estimate $SE(\bar{Y})$?
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When performing a paired t-test, what is typically being compared?
When performing a paired t-test, what is typically being compared?
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What adjustment is made to the standard deviation in the t-ratio when it is unknown?
What adjustment is made to the standard deviation in the t-ratio when it is unknown?
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Which of the following is NOT a type of t-test mentioned?
Which of the following is NOT a type of t-test mentioned?
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What is the purpose of log-transformation in data analysis?
What is the purpose of log-transformation in data analysis?
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What is the null hypothesis for a paired t-test in the context provided?
What is the null hypothesis for a paired t-test in the context provided?
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What does a rejection of the null hypothesis imply in this paired t-test?
What does a rejection of the null hypothesis imply in this paired t-test?
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What is the appropriate formula for the t-ratio in a paired t-test?
What is the appropriate formula for the t-ratio in a paired t-test?
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What is the purpose of calculating the p-value in a paired t-test?
What is the purpose of calculating the p-value in a paired t-test?
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How do you calculate the degrees of freedom in a paired t-test?
How do you calculate the degrees of freedom in a paired t-test?
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What is indicated by a p-value less than or equal to alpha (α) in a hypothesis test?
What is indicated by a p-value less than or equal to alpha (α) in a hypothesis test?
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Which step is NOT involved in conducting a paired t-test?
Which step is NOT involved in conducting a paired t-test?
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What does the symbol $d$ represent in the context of a paired t-test?
What does the symbol $d$ represent in the context of a paired t-test?
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What condition indicates that the t-ratio is calculated correctly for a one-sample t-test?
What condition indicates that the t-ratio is calculated correctly for a one-sample t-test?
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What must be true for the t distribution to approach the Z distribution?
What must be true for the t distribution to approach the Z distribution?
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In a two-tailed hypothesis test, when is the null hypothesis rejected?
In a two-tailed hypothesis test, when is the null hypothesis rejected?
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What is the formula to calculate the degrees of freedom in a one-sample t-test?
What is the formula to calculate the degrees of freedom in a one-sample t-test?
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Which of the following is NOT part of the steps to perform a one-sample t-test?
Which of the following is NOT part of the steps to perform a one-sample t-test?
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What does the term 'null hypothesis' represent in hypothesis testing?
What does the term 'null hypothesis' represent in hypothesis testing?
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Which test statistic is used for a right-tail one-sample t-test?
Which test statistic is used for a right-tail one-sample t-test?
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What is the effect of increasing the sample size on the standard error?
What is the effect of increasing the sample size on the standard error?
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What does the equation $Med\ Y_m = e^{(Z_m - Z_f)}$ indicate?
What does the equation $Med\ Y_m = e^{(Z_m - Z_f)}$ indicate?
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What is the result of taking the antilog of the difference $ln(Y_m) - ln(Y_f)$?
What is the result of taking the antilog of the difference $ln(Y_m) - ln(Y_f)$?
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How is the confidence interval for the median salary ratio derived?
How is the confidence interval for the median salary ratio derived?
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What does the expression $ln(Y)$ signify in this context?
What does the expression $ln(Y)$ signify in this context?
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Which of the following statements correctly interprets the results from the analysis?
Which of the following statements correctly interprets the results from the analysis?
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Why is the expression $ln(Y_m) ≠ ln(Y_f)$ significant?
Why is the expression $ln(Y_m) ≠ ln(Y_f)$ significant?
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What interpretation can be made from the confidence interval $(e^{0.0996}, e^{0.1942})$?
What interpretation can be made from the confidence interval $(e^{0.0996}, e^{0.1942})$?
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What does the term $Z_m$ represent in the analysis?
What does the term $Z_m$ represent in the analysis?
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Study Notes
Introduction
- Under the central limit theorem (CLT), the sample mean is approximately normally distributed.
- If the standard deviation of the population (σ) is unknown, we can estimate it using the sample standard deviation (s).
- The number of independent values used to estimate the standard error of the sample mean (SE(Y)) is called the degrees of freedom (ν).
t-ratio
- When the population standard deviation (σ) is known, we use the Z-ratio to make inferences about the population mean.
- If σ is unknown, we use the sample standard deviation (s) instead and call the ratio the t-ratio.
- The t-ratio follows a t-distribution with ν = n − 1 degrees of freedom.
- The t-distribution is symmetrical about its mean of zero, resembling a bell-shaped curve.
- As the degrees of freedom (ν) increase, the t-distribution approaches the Z-distribution.
One-Sample t-Test
- Used for hypothesis testing about the population mean (μ) when σ is unknown.
- Requires identifying the sample size (n), sample mean (Y), sample standard deviation (s), and the level of significance (α).
- Define null hypothesis (H0) and alternative hypothesis (H1) to determine the type of test (two-tailed, right-tailed, or left-tailed).
- Calculate the standard error of the sample mean (se) as s / √n.
- Calculate the test statistic (t-ratio) as (Y - μ0) / se, where μ0 is the hypothesized population mean.
- Calculate the degrees of freedom as ν = n - 1.
- Calculate the p-value based on the t-ratio and degrees of freedom, using appropriate formulas for two-tailed, right-tailed, or left-tailed tests.
- If the p-value is less than or equal to α, reject H0 and conclude that H1 is true. Otherwise, do not reject H0.
Paired t-Test
- Used for hypothesis testing about the difference between two related samples (e.g., pre-test and post-test scores).
- Define the mean difference as μd = μpost - μpre.
- Calculate the difference (d) for each observation, where d = Ypost - Ypre.
- Calculate the mean of d (d̄) and the standard deviation of d (sd).
- Calculate the standard error of the mean difference (se) as sd / √n.
- Calculate the test statistic (t-ratio) as d̄ / se.
- Calculate the degrees of freedom as ν = n - 1.
- Calculate the p-value based on the t-ratio and degrees of freedom.
- Reject H0 if the p-value is less than or equal to α.
Two-Sample t-Test
- Used for hypothesis testing about the difference between two independent samples (e.g., comparing the means of two groups).
- Two types: Pooled t-test and Welch t-test.
- Pooled t-Test: assumes equal variances in both groups.
- Welch t-Test: does not assume equal variances in both groups.
- Use the Welch t-test if the ratio of sample standard deviations (s1 / s2) is greater than or equal to 2.
Log Transformation
- Used to transform skewed datasets into symmetric ones.
- Most common choice is the natural logarithm (ln) transformation.
- Apply log transformation when data is skewed to the right (positively skewed) or when the spread is higher in the group with a larger center (median).
- Log-transformation aims to create two symmetric samples with similar spreads but potentially different centers.
Logged Data: Observational Studies
- After log transformation, if the data is symmetric, then the mean and median of the log-transformed data are equal: Mean[ln(Y)] = Median[ln(Y)].
- The log transformation preserves order: Median[ln(Y)] = ln[Median Y].
- By combining these equations, we can estimate the ratio of medians: exp(Z̄m - Z̄f) ≈ (Med Ym) / (Med Yf).
- The antilog of the difference in means (Z̄m - Z̄f) estimates the ratio of medians in the original data.
Example – Salary Discrimination
- Log transformation was applied to salary data to address skewness and facilitate analysis.
- The difference in log-transformed salaries for males (Zm) and females (Zf) provided an estimate of the ratio of median salaries: exp(Zm̄ - Zf̄) ≈ (Med Ym) / (Med Yf).
- In the example, the median salary for males was estimated to be 15.83% more than the median salary for females.
- The 95% confidence interval (CI) for the ratio of medians was also calculated, providing a range of plausible values for the difference in median salaries.
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Description
This quiz covers key concepts related to the central limit theorem, t-ratios, and one-sample t-tests. It is essential for understanding hypothesis testing when the population standard deviation is unknown. Test your knowledge on these fundamental statistical principles!