A producer is very proud of the peaches he grows. He claims that, on average, 10% or less of his
peaches in a bushel are defective. You take a sample of 400 peaches and find that 50 are
defective. If α = 0.025, what will be the decision?
To determine the decision in this scenario, we need to conduct a hypothesis test using the given information.
Step 1: State the null and alternative hypotheses:
- Null Hypothesis (H₀): The proportion of defective peaches is equal to or less than 10% (p ≤ 0.10)
- Alternative Hypothesis (H₁): The proportion of defective peaches is greater than 10% (p > 0.10)
Step 2: Determine the level of significance (α):
The level of significance (α) is given as 0.025.
Step 3: Calculate the test statistic and p-value:
To determine the test statistic, we can use the Z-test for proportions. The formula to calculate the test statistic is:
Z = (p̂ - p₀) / √[(p₀(1 - p₀)) / n]
Where:
p̂ is the sample proportion (defective peaches in the sample divided by the total sample size),
p₀ is the hypothesized proportion, which is 0.10 in this case,
n is the sample size.
In this case, p̂ = 50/400 = 0.125, p₀ = 0.10, and n = 400.
Plugging these values into the formula gives:
Z = (0.125 - 0.10) / √[(0.10(1 - 0.10)) / 400]
Z = (0.025) / √[0.09 / 400]
Z ≈ 0.025 / √0.000225
Z ≈ 0.025 / 0.015
Using a statistical calculator or table, we find that the test statistic is approximately 1.67.
Next, we need to calculate the p-value associated with this test statistic. Since the alternative hypothesis is one-sided (stating "greater than 10%"), we need to find the probability of obtaining a test statistic as extreme as the one observed or more extreme (i.e., to the right of the calculated Z-value).
For a Z-value of 1.67, the corresponding p-value is approximately 0.0475 (or 4.75%).
Step 4: Make a decision:
To make a decision, we compare the p-value to the level of significance (α).
Since the p-value (0.0475) is less than the level of significance (0.025), we reject the null hypothesis.
Therefore, based on the given information and statistical analysis, the decision is to reject the producer's claim that the proportion of defective peaches is 10% or less.