Welcome back to the AI Bayeslab Statistics series. I recognize that data is plentiful in your work. With the AI Agent online tool, you’ll find that almost 90% of your data analysis tasks can be managed independently, without requiring a dedicated data analyst.

We established the scenario in which σ₁² is equal to σ₂². Previously, both populations exhibited normal distributions. Today, we will explore a different common data condition: when σ₁² is not equal to σ₂².

What is the equation for hypothesis testing when σ₁² ≠ σ₂²?

So when the data sample accords with the condition as mentioned above, we can use a hypothesis with a t-statistic; the formula is as follows:

Also, as long as sampling involves a large sample size, defined as having more than 30 individuals per sample group. We can use a hypothesis with a Z-statistic. The formula is as follows:

What homogeneity of variance?

In hypothesis testing, homogeneity of variance assumes that various groups or samples exhibit equal variances.

This assumption is commonly necessary in analyses such as Analysis of Variance (ANOVA).

• When variances are equal, it is termed “homogeneity.”

• Meanwhile, unequal variances are called “inhomogeneity” or “heteroscedasticity.”

Assuming homogeneity when variances are actually inhomogeneous can result in misleading statistical test outcomes, heightening the likelihood of Type I or Type II errors.

Thus, it is crucial to verify the homogeneity of variances using tests like Levene’s or Bartlett’s tests before proceeding with further statistical analyses. If inhomogeneity is present, one might consider employing robust statistical methods or performing data transformations to lessen its effects.

Example: Two Population Means with Unknown Standard Deviations (σ₁², σ₂² unknown, and heterogeneous)

Data sample description_ Reaction time:

- Group1: male-student.csv — Sample size: 45, Reaction time X̅₁,= 705.400, the variance of a sample S₁² = 406.151

- Group2: female-student.csv — Sample size: 36, Reaction time X̅₂ = 732.80, the variance of a sample S₂² = 898.179

Question: Does the data demonstrate that men react significantly faster than women?

There is considerable disparity between S₁² = 406.151 and S₂² = 898.179, leading us to classify them as heterogeneous or inhomogeneous.

This is a case of two populations with Unknown Standard Deviations, too, so let’s analyze it step by step as we always do.

Step1. State the Hypothesis

• Null Hypothesis (H₀):

• Typically represents no significant difference or effect. For example,

• H₀: μ = 100 (the population mean equals 100).

• Alternative Hypothesis (H₁):

• Indicates a significant difference or effect.

• For example, H₁: μ ≠ 100 (the population mean does not equal 100).

In this scenario, we use a left-tailed test. Null Hypothesis (H₀) vs Alternative Hypothesis (H₁)

H₀: μ₁ — μ₂ ≥0 or μ₁ ≥ μ₂

H₁: μ₁ — μ₂ < 0 or μ₁ < μ₂

Step2. Choose the Significance Level and Appropriate Test

The population variances are both unknown and adhere to a t-distribution curve. To recap, we outlined four criteria that help choose a suitable hypothesis test. We developed a summary sheet to assess both the sample and the population, aiding us in choosing the test statistic and its respective formula.

In this case, we can apply the Z-distribution and the test statistic outlined below. We can also use the Z-statistic when our sample sizes are greater than 30, but t’ can also be used to calculate the degrees of freedom.

Step3. Calculate the Test Statistic

We have calculated the means of the two samples above:

▶︎ Male: X̅₁ = 732.80, 𝑛₁ =45, S₁² = 406.151

▶︎ Female: X̅₂ = 732.80, 𝑛₂ = 36, S₂² = 898.179

We can calculate the corresponding t value by substituting these into the formula.

However, at this point, we will not perform the calculation manually. We will integrate it into one step and let the AI Agent calculate based on the selected formula and the null hypothesis.

Step4. Make a Decision

• t’ value: -4.700715396104711

• df’ value: 58.78387470943872

• p-value: 8.045481968498432e-06

• Reject the null hypothesis: Men react significantly faster than women.

AI prompts serve as potent tools that help us efficiently complete various tasks and achieve desired outcomes. By utilizing technology in this manner, I believe we have a remarkable opportunity to reintroduce statistics into everyday life, reigniting interest in data and fostering analytical thinking.

Reviving this is crucial as statistics offer significant insights into different life aspects, aiding individuals in making better-informed choices. By ensuring statistics are accessible and pertinent, we improve comprehension and enable individuals to interact meaningfully with data, ultimately enhancing their daily lives.

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