Statistics: t-Tests and Central Limit Theorem

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Questions and Answers

What is the appropriate condition to use a Welch t-test instead of a pooled t-test?

The variances of the two samples are assumed to be equal.

The assumption is that one standard deviation is greater than the other. (correct)

The ratio of the standard deviations must be less than 2.

The sample sizes must be equal.

When should log transformation be considered for datasets?

When the number of samples is very high.

When the standard deviation is uniform across groups.

Only in datasets that are perfectly symmetrical.

In one-sample tests with positively skewed data. (correct)

What does the ideal result of a log transformation include?

Data that maintains its original distribution.

A significant decrease in all sample sizes.

Two symmetric samples with varying centers and similar spreads. (correct)

Symmetric samples with different spreads and similar centers.

How is the standard error for the samples’ mean difference calculated in a Welch t-test?

Using variances and the degrees of freedom as min 𝑛1 − 1, 𝑛2 − 1. Signup and view all the answers

What occurs when the log-transformed data are symmetric?

The mean and median are equal. Signup and view all the answers

What is the purpose of using the t-ratio instead of the z-ratio?

To make inferences when the population standard deviation is unknown Signup and view all the answers

In the formula for the standard error of the sample mean, $SE(\bar{Y})$, what does 'n' represent?

The number of observations in the sample Signup and view all the answers

Under the Central Limit Theorem (CLT), which distribution is the sample mean approximately normally distributed?

Sampling distribution of the sample mean Signup and view all the answers

What is the term for the number of independent values used to estimate $SE(\bar{Y})$?

Degrees of freedom Signup and view all the answers

When performing a paired t-test, what is typically being compared?

Two related samples or matched samples Signup and view all the answers

What adjustment is made to the standard deviation in the t-ratio when it is unknown?

It is replaced with the sample standard deviation Signup and view all the answers

Which of the following is NOT a type of t-test mentioned?

Independent t-test Signup and view all the answers

What is the purpose of log-transformation in data analysis?

To stabilize variance and make a distribution more normal Signup and view all the answers

What is the null hypothesis for a paired t-test in the context provided?

The mean difference in scores is equal to 0. Signup and view all the answers

What does a rejection of the null hypothesis imply in this paired t-test?

The training module, on average, increased the test scores. Signup and view all the answers

What is the appropriate formula for the t-ratio in a paired t-test?

$t-ratio = \frac{d̄}{s_{d} / \sqrt{n}}$ Signup and view all the answers

What is the purpose of calculating the p-value in a paired t-test?

To help decide whether to reject the null hypothesis. Signup and view all the answers

How do you calculate the degrees of freedom in a paired t-test?

By subtracting 1 from the number of observations. Signup and view all the answers

What is indicated by a p-value less than or equal to alpha (α) in a hypothesis test?

You reject the null hypothesis. Signup and view all the answers

Which step is NOT involved in conducting a paired t-test?

Calculate the standard deviation of the dataset. Signup and view all the answers

What does the symbol $d$ represent in the context of a paired t-test?

The mean difference between paired observations. Signup and view all the answers

What condition indicates that the t-ratio is calculated correctly for a one-sample t-test?

The standard error is computed using $s/n$. Signup and view all the answers

What must be true for the t distribution to approach the Z distribution?

The degrees of freedom must increase. Signup and view all the answers

In a two-tailed hypothesis test, when is the null hypothesis rejected?

When the p-value is less than or equal to the level of significance. Signup and view all the answers

What is the formula to calculate the degrees of freedom in a one-sample t-test?

$n - 1$ Signup and view all the answers

Which of the following is NOT part of the steps to perform a one-sample t-test?

Calculate the correlation coefficient. Signup and view all the answers

What does the term 'null hypothesis' represent in hypothesis testing?

A statement asserting no effect or no difference exists. Signup and view all the answers

Which test statistic is used for a right-tail one-sample t-test?

$t-ratio = rac{Ȳ - μ_0}{s_e}$ Signup and view all the answers

What is the effect of increasing the sample size on the standard error?

The standard error decreases. Signup and view all the answers

What does the equation $Med\ Y_m = e^{(Z_m - Z_f)}$ indicate?

The median salary for males is estimated to be 15.83% more than that for females. Signup and view all the answers

What is the result of taking the antilog of the difference $ln(Y_m) - ln(Y_f)$?

It estimates the ratio of the median salaries between males and females. Signup and view all the answers

How is the confidence interval for the median salary ratio derived?

It is obtained by exponentiating the lower and upper bounds of $Z_m - Z_f$. Signup and view all the answers

What does the expression $ln(Y)$ signify in this context?

The natural logarithm of the population average salary. Signup and view all the answers

Which of the following statements correctly interprets the results from the analysis?

The confidence interval indicates that males earn, on average, more than females. Signup and view all the answers

Why is the expression $ln(Y_m) ≠ ln(Y_f)$ significant?

It suggests that the logarithmic transformation maintains the order of the data. Signup and view all the answers

What interpretation can be made from the confidence interval $(e^{0.0996}, e^{0.1942})$?

The median salary for males might be between 1.11 and 1.21 times that of females. Signup and view all the answers

What does the term $Z_m$ represent in the analysis?

The logarithmic transformation of the median salary of males. Signup and view all the answers

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!