Real Time Scenario for Two Independent sample Z test for Equal and Unequal test with Data

Certainly! Let’s consider a real-world scenario where we would use a two independent sample $z$ test:

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Scenario: Energy Drinks and Reaction Times

Imagine you work at a research institution and you’re tasked with investigating whether two different brands of energy drinks (let’s call them Brand A and Brand B) have any effect on the reaction times of individuals.

You decide to set up an experiment where:

100 participants drink Brand A.
Another 100 participants drink Brand B.
After drinking, you measure their reaction times using a standard test.

Hypotheses:

Null Hypothesis ( $H_{0}$ ): The mean reaction time of individuals drinking Brand A is equal to the mean reaction time of individuals drinking Brand B.
Alternative Hypothesis ( $H_{1}$ ): The mean reaction time of individuals drinking Brand A is not equal to the mean reaction time of individuals drinking Brand B.

Data:

Let’s assume you’ve gathered the following data:

Brand A:
- Sample Size ( $n_{1}$ ): 100
- Sample Mean ( $x ˉ_{1}$ ): 420 milliseconds (ms)
- Sample Standard Deviation ( $s_{1}$ ): 50 ms
Brand B:
- Sample Size ( $n_{2}$ ): 100
- Sample Mean ( $x ˉ_{2}$ ): 430 ms
- Sample Standard Deviation ( $s_{2}$ ): 55 ms

Two Independent Sample $z$ Test:

We can use the formula for the test statistic for two independent samples:

Where:

$x ˉ_{1}$ and $x ˉ_{2}$ are the sample means of the two groups.
$s_{1}$ and $s_{2}$ are the standard deviations of the two groups.
$n_{1}$ and $n_{2}$ are the sample sizes of the two groups.

Let’s calculate the $z$ test statistic using the data provided.

# Given data
n1, n2 = 100, 100  # sample sizes
x1_bar, x2_bar = 420, 430  # sample means (in milliseconds)
s1, s2 = 50, 55  # sample standard deviations (in milliseconds)

# Calculating the z test statistic
z = (x1_bar - x2_bar) / ((s1**2 / n1 + s2**2 / n2) ** 0.5)
z

RESULT

-1.3453455879926248

The calculated $z$ test statistic is approximately $- 1.345$ .

Now, to determine if this difference is statistically significant, you would compare the calculated $z$ value to a critical $z$ value (for a given level of significance, usually $α = 0.05$ for a two-tailed test). If the calculated $z$ value is greater than the critical value in magnitude, you would reject the null hypothesis.

Assuming a significance level of $α = 0.05$ for a two-tailed test, the critical $z$ values are approximately $\pm 1.96$ . Since the magnitude of our calculated $z$ value ( $∣ - 1.345∣$ ) is less than 1.96, we would fail to reject the null hypothesis.

This means that, based on our sample data, there isn’t enough evidence to suggest that there’s a significant difference in reaction times between individuals who drank Brand A and those who drank Brand B.

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Scenario: Energy Drinks and Reaction Times

Hypotheses:

Data:

Two Independent Sample $z$ Test:

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