9.4 Significance Tests For a Proportion

Hypothesis Testing Overview

• Introduction to Hypothesis Testing
    - Focus on performing a full hypothesis test for practice due to upcoming quizzes.
    - Understanding the difference between one-tailed and two-tailed tests.

Context of the Hypothesis Test

• Scenario:
    - A news report claims that 75% of restaurant employees feel that work stress has a negative impact on their personal lives.
    - Managers at a restaurant chain want to assess if this claim holds true among their employees.

• Data Collection:
    - A random sample of 100 employees (denoted as $n = 100$) was taken.
    - Out of these, 68 employees reported that work stress negatively impacts their personal lives.

Parameters and Hypotheses

• Parameters of Interest:
    - Population proportion denoted as $p$.
    - Null hypothesis ($H_0$): $p = 0.75$ (claim from the news report).
    - Alternative hypothesis ($H_a$): $p <br>eq 0.75$ (to test if the proportion differs from 75%).

• Significance Level:
    - The significance level ($ext{alpha}$) set for testing is 10% (0.10).
    - Common assumption if not specified is $ext{alpha} = 0.05$.

Calculating the Sample Proportion

• Sample Proportion ($p̂$):
    - Calculated as the number of "yes" responses divided by the sample size:
      $p̂ = rac{68}{100} = 0.68$.

Testing Procedure

• Type of Test:
    - This test is identified as a one-sample proportion Z-test.

Conditions for the Test

1. Randomness:
      - The sample should be random, as indicated in the problem statement (random sample of 100).

2. Normal Distribution:
      - Check conditions:
        - $n imes p > 10$ and $n imes (1 - p) > 10$,
        - Calculate:
          - $n imes 0.75 = 100 imes 0.75 = 75$ (satisfied),
          - $n imes (1 - 0.75) = 100 imes 0.25 = 25$ (satisfied).
      - Satisfied both conditions: Therefore, the sampling distribution can be approximated as normal.

Mean and Standard Deviation

• Mean of the Sampling Distribution:
    - $ext{Mean (μ)} = p = 0.75$.

• Standard Deviation:
    - Formula for standard deviation of the sample proportion:
      $ext{SD} = rac{ ext{sqrt}(p(1 - p))}{ ext{sqrt}(n)} = ext{sqrt}\bigg( rac{0.75 imes 0.25}{100} \bigg)$.

Z-Score Calculation

• Z-Score Formula:
    - $Z = rac{p̂ - p}{ ext{SD}}$.
    - Plugging in values:
      - $Z = rac{0.68 - 0.75}{ ext{SD}}$ (calculated standard deviation needed).
    - Example Z-Score Calculation:
      - Assuming SD calculated is approximately 0.043.
      - $Z = rac{-0.07}{0.043} ext{ (calculated Z value)}.$

Finding the P-value

• Reference the Z-Score table to find the corresponding area.
    - Example Z-Score ($-1.63$) yields a one-tailed p-value of: $P(Z < -1.63) = 0.0516$.

• Two-tailed Test: Calculating P-Value:
    - Since it is a two-tailed test, double the one-tailed p-value:
      $ext{P-value} = 2 imes 0.0516 = 0.1032$.

Conclusion of the Hypothesis Test

• Compare P-value to Alpha:
    - Given $ext{P-value} = 0.1032$ and $ext{Alpha} = 0.10$.
    - Since $P > ext{alpha}$, we Fail to Reject the null hypothesis.

Interpret Results:

• Conclusion statement: "We do not have sufficient evidence to suggest that the proportion of employees who feel work stress affects their personal life differs from 0.75."
    - This indicates that the sample does not provide enough evidence to dispute the claim made in the news report.

Summary and Review

• Keep in mind:
    - The process involves carefully stating hypotheses, checking test conditions, calculating Z-scores and p-values, and comparing them to establish conclusions.

• Upcoming quizzes will include similar questions based on the details discussed in this session.