Study Notes on Statistical Inference - Population Proportion

Statistical Inference - Population Proportion

Test of Significance for a Proportion

Introduction

  • The primary focus of these notes is on the statistical inference for population proportions.

  • A key aspect of this involves executing a Test of Significance for a Proportion.

Four Steps in Carrying Out a Significance Test

  1. State the null and alternative hypotheses.

  2. Check conditions and then calculate the test statistic.

  3. Find the P-value using the appropriate distribution.

  4. Make a decision and state your conclusion in the context of the specific setting of the test.

    • This structure is essential for conducting a proper significance test.

    • This systematized approach is endorsed for all tests in statistical inference.

Examples of Significance Tests

  1. Parliamentarian Vote Example:

    • Null Hypothesis (H0): $p = 0.5$ (The proportion of constituents favoring the proposal is 50%).

    • Alternative Hypothesis (Hₐ): $p > 0.5$ (More than 50% of constituents favor the proposal).

  2. Pharmaceutical Company Claim:

    • Null Hypothesis (H0): $p = 0.2$ (The proportion of patients experiencing side effects is 20%).

    • Alternative Hypothesis (Hₐ): $p < 0.2$ (Less than 20% of patients experience side effects).

  3. Children Raised by Grandparents:

    • Null Hypothesis (H0): $p = 0.05$ (5% of children are raised by grandparents).

    • Alternative Hypothesis (Hₐ): $p
      eq 0.05$ (The proportion has changed from 5%).

Step 1: State the Hypotheses

  • Null hypothesis (H0): $p = p0$ (Where $p0$ is the null value.).

  • Alternative hypothesis (Hₐ): There are three forms depending on the direction:

    • One-sided, right-sided: $Hₐ: p > p_0$

    • One-sided, left-sided: $Hₐ: p < p_0$

    • Two-sided: $Hₐ: p
      eq p_0$

Step 2: Condition Check
Step 2a: Check Conditions

  1. Random Sample Requirement:

    • The sample should ideally be a random sample from the population.

    • While a random sample is preferred, it is acceptable if the sample is representative of the population relevant to the question.

  2. Sample Size Requirement:

    • The sample size must be sufficient to ensure the approximate normality of the sampling distribution.

    • Specifically, check if both $np0$ and $n(1 - p0)$ are at least 10 to confirm this condition.

Step 3: Test Statistic and P-value
Step 3: P-value

  • The P-value is calculated to determine the strength of the test results.

Step 4: Decision and Conclusion

  • The decision regarding the null hypothesis is based on the P-value and the pre-defined level of significance, denoted by $$ (alpha).

  • Decision Rules:

    • If the P-value $ ext{≤} $, Reject the null hypothesis, concluding there is statistically significant evidence for the alternative hypothesis.

    • If the P-value $>$, Cannot reject the null hypothesis, concluding that there is insufficient evidence to support the alternative hypothesis.

Example: Mendel’s Peas

  • Context: Pure bred peas were crossed, resulting in smooth and wrinkled peas. The first generation (F1) was all smooth. In the second generation (F2), the resulting counts were:

    • Smooth Peas: 5474

    • Wrinkled Peas: 1850

  • The hypothesis tested was whether this data supports the 75% dominant trait occurrence.

  • Test Structure:

    • Significance level: $ = 0.05$.

    • Hypotheses:

      • Null Hypothesis (H0): Proportion of dominant trait = 0.75.

      • Alternative Hypothesis (Hₐ): Proportion of dominant trait $
        eq 0.75$.

  • Decision:

    • With a calculated P-value of 0.610, which is greater than $ = 0.05$, we fail to reject H0.

    • Conclusion: There is no significant evidence, at the 5% level of significance, that the proportion of the dominant (smooth) trait occurring in F2 is different from 75%. Thus, data supports H0 and aligns with the conclusion of a 75% dominant trait occurrence in F2.

Effect of Sample Size on Statistical Significance

  • Observations based on different sample sizes:

    • Sample 1: Fails to reject H0.

    • Sample 2: Rejects H0.

  • Conclusion: As sample size increases, the likelihood of rejecting the null hypothesis becomes higher.

Conditions Summary

  • To validate tests conducted on proportions, the following conditions must be checked:

    • A random sample is needed.

    • Only use the sample data (no plus four method).

    • Z-test validity is confirmed if both $np$ and $n(1 - p)$ are at least 10.