Data Science Interview Prep: Q60

P-Value in Hypothesis Testing. (Category: Statistics)

Explain p-value in layman’s terms.

Solution:

Kindly read the ‘Z-test versus T-test: When and How to Use Them in Hypothesis Testing’ newsletter before reading the current newsletter.

Link: Z-test versus T-test: When and How to Use Them in Hypothesis Testing

In the previous newsletter, we examined the conditions under which the Z-test and t-test are applicable. In this newsletter, we delve into the concept of the p-value and demonstrate how it is computed for both t and Z test statistics.

The p-value is the probability, under the assumption that the null hypothesis (H₀) is true, of obtaining a test statistic (or result) at least as extreme as the one actually observed.

In hypothesis testing, we start with:

• Null hypothesis (H₀): A statement of no effect or no difference.

• Alternative hypothesis (H_a): What you want to test - could be one-sided or two-sided.

The p-value shows how consistent your data are with H₀. A smaller p-value means stronger evidence against H₀.

• If the p-value is less than the chosen significance level (commonly α = 0.05), we reject the null hypothesis.

• If the p-value is greater than α, we fail to reject the null hypothesis, indicating insufficient evidence to support an alternative hypothesis.

In hypothesis testing, a parameter is a numerical characteristic that describes a population — for example:

• the population mean (μ)

• the population proportion (p)

• the population variance (σ²)

Let us try to understand, step by step, how we calculate the p-value for a given hypothesis.

Let us explore and understand the above concepts with some numerical problems.

This figure displayed above for the one-tailed Z-test for battery life can be summarized as follows:

• The black curve represents the standard normal distribution under the null hypothesis H₀.

• The light blue region is where we fail to reject H₀.

• The light red region is the rejection region (α = 0.05), where results are unlikely under H₀.

• The dark red shaded area shows the observed p-value region for the test statistic Z = 2.88.

• The blue dashed line marks the observed Z-value, and the red dotted line shows the critical Z-value at 1.64.

Since the observed Z-value falls in the rejection region, there is strong evidence to support the claim that the average battery life is greater than 100 hours.

This figure displayed above for the two-tailed t-test for the coffee shop wait time can be summarized as follows:

• The black curve represents the t-distribution under the null hypothesis (H₀), with 11 degrees of freedom.

• The light blue region indicates where we fail to reject H₀.

• The light red regions in both tails represent the rejection regions (α = 0.05, with α/2 = 0.025 in each tail), where results are unlikely under H₀.

• The dark red shaded areas show the observed p-value regions for the test statistic t = 2.31, mirrored in both tails since this is a two-tailed test.

• The blue dashed line marks the observed t-value, and the red dotted lines show the critical t-values at ±2.201.

Since the observed t-value falls within the rejection region, there is sufficient evidence to conclude that the average wait time differs significantly from 3 minutes.

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Reference:

• P-Value: https://en.wikipedia.org/wiki/P-value