Question

Identify the null hypothesis, alternative hypothesis, test statistic, p-value, decision about the null hypothesis, and final conclusion that addresses the original claim. round the test statistic to two decimal places and the p-value to four decimal places.

a supplier of digital memory cards claims that no more than 1% of the cards are defective. in a random sample of 600 memory cards, it is found that 3% are defective, but the supplier claims that this is only a sample fluctuation. at the 0.01 level of significance, test the supplier's claim that no more than 1% are defective.

Answer 1

The null hypothesis, stating that no more than 1% of memory cards are defective, is rejected in favor of the alternative hypothesis that over 1% are defective due to a low p-value compared to the 0.01 significance level.

Step-by-step explanation:

Hypothesis Testing for a Sample Proportion

To address the question regarding a supplier's claim about the defect rate in digital memory cards, we first need to establish the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis reflects the supplier's claim while the alternative hypothesis states what we suspect might be the case if the claim is not true.

• A. Null Hypothesis (H0): The proportion of defective memory cards is no more than 1% (p ≤ 0.01).
• B. Alternative Hypothesis (Ha): The proportion of defective memory cards is greater than 1% (p > 0.01).

The random variable, P', represents the sample proportion of defective memory cards observed in the sample. In this case, P' = 3% or 0.03.

Calculation of the Test Statistic

The test statistic is calculated using the sample proportion, the hypothesized proportion, and the sample size. Using the z-test for proportions:

• C. Test Statistic (z): (P' - P) / sqrt[(P(1 - P) / n)]

Substituting P' = 0.03, P = 0.01, and n = 600:

z = (0.03 - 0.01) / sqrt[(0.01(1 - 0.01) / 600)]

After calculations z ≈ 8.16 (rounded to two decimal places).

Determination of the P-value

The p-value corresponds to the probability of observing a statistic as extreme as the test statistic under the null hypothesis. Given that the test statistic z is approximately 8.16, the p-value is significantly less than 0.01, indicating a very rare event under the null hypothesis.

• D. P-value: Very small (less than 0.0001)

Decision and Conclusion

Comparing the p-value to the significance level (α = 0.01), and because the p-value < α, we reject the null hypothesis. There is sufficient evidence at the 1 percent level of significance to conclude that more than 1% of the memory cards are defective, contrary to the supplier's claim.

E. Type I Error: Rejecting H0 when it is true (falsely declaring a defect rate higher than 1%).

F. Type II Error: Failing to reject H0 when it is false (not detecting a defect rate higher than 1% when it indeed is).