One Proportion Z-Test

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Number of Successes:

Number of Trials:

Hypothesized Proportion:

Significance Level:

Alternative Hypothesis:

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One-Proportion Z-Test

Definition

One-Proportion Z-Test is used to test if a population proportion is significantly different from a hypothesized value. It's commonly used for analyzing success rates, percentages, or proportions in a single group.

Formula

Test Statistic:

z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}

Where:

$\hat{p}$ = sample proportion
$p_0$ = hypothesized population proportion
$n$ = sample size

Confidence Interval:

\text{Two-sided: }CI = \hat{p} \pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

\text{One-sided: }CI = \hat{p} + z_{\alpha}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Where $z_{\alpha/2}$ and $z_{\alpha}$ are the critical values for the desired confidence level.

Key Assumptions

Random Sample: Data must be a random sample from the population

Independence: Observations must be independent

Sample Size: Both np₀ and n(1-p₀) should be ≥ 10

Common Pitfalls

Using the test when sample size conditions aren't met
Not checking for independence of observations
Confusing statistical significance with practical importance

Practical Example

Testing if a new manufacturing process meets quality standards:

Step 1: State the Data

Sample size: $n$ = 100 units
Defect-free units: $x$ = 45 units
Sample proportion: $\hat p = 45/100 = 0.45$

Step 2: State Hypotheses

$H_0: p = 0.50$ (standard quality level)
$H_a: p \neq 0.50$ (different from standard)
$\alpha = 0.05$

Step 3: Calculate Test Statistic

Z-statistic:

z = \frac{0.45 - 0.50}{\sqrt{\frac{0.50(1-0.50)}{100}}} = -1.00

Step 4: Calculate P-value

For two-tailed test:

p\text{-value} = 2(1 - \Phi(|z|)) = 2(1 - \Phi(1.00)) = 0.317

Step 5: Calculate Confidence Interval

\text{CI: } 0.45 \pm 1.96\sqrt{\frac{0.45(1-0.45)}{100}} = [0.352, 0.548]

Step 6: Draw Conclusion

Critical value at 5% significance level:

z_{\alpha/2} = 1.96

Since $|z| < z_{\alpha/2}$ and $p\text{-value} > 0.05$ , we fail to reject $H_0$ . There is insufficient evidence to conclude that the process quality differs from the standard level.

Effect Size

Cohen's h for one proportion:

h = 2\arcsin\sqrt{\hat{p}} - 2\arcsin\sqrt{p_0}

Interpretation:

Small effect: $|h| \approx 0.2$
Medium effect: $|h| \approx 0.5$
Large effect: $|h| \approx 0.8$

Power Analysis

Required sample size for desired power:

n = \frac{(z_{1-\alpha/2} + z_{1-\beta})^2 p_0(1-p_0)}{(p_a-p_0)^2}

Where:

$p_a$ = alternative proportion
$\beta$ = Type II error rate
$\alpha$ = significance level

Code Examples

1x <- 45    # Number of successes
2n <- 100   # Number of trials
3p <- 0.5   # Null hypothesis proportion
4
5# Perform one-proportion test
6result <- prop.test(
7  x = x,
8  n = n,
9  p = p,
10  alternative = "two.sided",
11  correct = FALSE  # No continuity correction
12)
13
14# Print results
15print(result)

Python

1from statsmodels.stats.proportion import proportions_ztest
2import numpy as np
3from scipy import stats
4
5# Example data
6p0 = 0.5
7success1, n1 = 45, 100
8p1 = success1 / n1
9
10# Perform one-proportion z-test
11zstat, pvalue = proportions_ztest(
12    success1, n1, value=p0, alternative='two-sided'
13)
14
15print(f'z-statistic: {zstat:.4f}')
16print(f'p-value: {pvalue:.4f}')
17
18# construct confidence interval
19alpha = 0.05
20z_critical = stats.norm.ppf(1 - alpha / 2)
21margin_of_error = z_critical * np.sqrt((p1 * (1 - p1) / n1))
22prop_diff = p1
23ci_lower = prop_diff - margin_of_error
24ci_upper = prop_diff + margin_of_error
25print(f'Confidence interval: ({ci_lower:.4f}, {ci_upper:.4f})')

Alternative Tests

Consider these alternatives when assumptions are violated:

Exact Binomial Test: For small samples
Chi-square Goodness of Fit: For categorical data
Bootstrap Methods: When sample size conditions aren't met

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