Worms in Apples (One Sample Proportion Test)

Suppose we want to test if the proportion of apples with worms in an orchard is different from the commonly accepted proportion of 5%. We will use a one-sample proportion test to determine if there is a significant difference.

Step 1: State the Hypotheses

The null hypothesis and the alternative hypothesis can be stated as follows:


where is the true proportion of apples with worms.

Step 2: Collect Data

Suppose we randomly sample 150 apples from the orchard and find that 12 of them have worms. We can summarize this information as follows:

Step 3: Calculate the Test Statistic

The test statistic for a one-sample proportion test is calculated using the formula:

where is the null hypothesis proportion (0.05 in this case).

Step 4: Determine the Critical Value

Using a standard normal distribution (z-distribution), we find the critical value for a two-tailed test at the significance level is approximately .

Step 5: Make a Decision

Since the calculated -statistic (1.48) is less than the critical value (1.96), we fail to reject the null hypothesis.

Conclusion

At the significance level, we do not have enough evidence to conclude that the proportion of apples with worms is different from 5%. Therefore, we fail to reject the null hypothesis that the proportion of apples with worms in the orchard is 5%.

implementation

Python

import numpy as np
from statsmodels.stats.proportion import proportions_ztest

# Data
n = 150  # total number of apples sampled
x = 12   # number of apples with worms
p0 = 0.05  # null hypothesis proportion

# Sample proportion
p_hat = x / n

# Perform one-sample proportion z-test
z_statistic, p_value = proportions_ztest(count=x, nobs=n, value=p0)

# Print results
print(f"Sample Proportion: {p_hat}")
print(f"Z-Statistic: {z_statistic}")
print(f"P-Value: {p_value}")

# Conclusion based on alpha = 0.05
alpha = 0.05
if p_value < alpha:
    print("Reject the null hypothesis.")
else:
    print("Fail to reject the null hypothesis.")

R

# Data
n <- 150  # total number of apples sampled
x <- 12   # number of apples with worms
p0 <- 0.05  # null hypothesis proportion

# Sample proportion
p_hat <- x / n

# Perform one-sample proportion z-test
z_statistic <- (p_hat - p0) / sqrt(p0 * (1 - p0) / n)
p_value <- 2 * (1 - pnorm(abs(z_statistic)))  # two-tailed test

# Print results
cat("Sample Proportion:", p_hat, "\n")
cat("Z-Statistic:", z_statistic, "\n")
cat("P-Value:", p_value, "\n")

# Conclusion based on alpha = 0.05
alpha <- 0.05
if (p_value < alpha) {
    cat("Reject the null hypothesis.\n")
} else {
    cat("Fail to reject the null hypothesis.\n")
}