Company A makes a large shipment to Company B. Company B can reject the shipment if they can conclude that the proportion of defective items in the shipment is larger than 0.1. In a sample of 400 items from the shipment, Company B finds that 59 are defective. Conduct the appropriate hypothesis test for Company B using a 0.05 level of significance.

To test whether the proportion of defective items in the shipment exceeds 0.1, Company B can use a hypothesis test for the proportion. Here's how to perform the test:

Step 1: State the null and alternative hypothesis.

- The null hypothesis (H0): p = 0.1, where p is the true proportion of defective items in the shipment. - The alternative hypothesis (H1): p > 0.1, since Company B is concerned with the proportion being greater than 10%.

Step 2: Choose the significance level.

- The significance level is given as α = 0.05.

Step 3: Determine the test statistic.

For large samples (n*p and n*(1-p) both greater than 5), the test statistic for a proportion is a z-score, which is calculated using the formula:

z = (p̂ - p0) / √(p0(1 - p0) / n)

In this case:

- n = 400 (size of the sample) - p̂ = 59/400 = 0.1475 (sample proportion of defective items) - p0 = 0.1 (claimed proportion of defective items under the null hypothesis)

z = (0.1475 - 0.1) / √(0.1(1 - 0.1) / 400) z = 0.0475 / √(0.1*0.9/400) z = 0.0475 / √0.000225 z = 0.0475 / 0.015 z ≈ 3.17

Step 4: Calculate the p-value or compare the z-score to the critical value.

Since we are dealing with a right-tail test, we will look for the p-value corresponding to our z-score of 3.17. In a standard normal distribution (z-distribution), a z-score of 3.17 has a very small p-value.

Alternatively, we can find the critical z-value at α = 0.05 for a right-tailed test. Looking up the z-table or using statistical software, we find that the critical z-value is approximately 1.645. Our calculated z-score (3.17) is greater than the critical z-value.

Step 5: Make a decision.

Based on the p-value being lower than the significance level or our z-score being higher than the critical value, we reject the null hypothesis.

Conclusion: There is enough evidence at the 0.05 significance level to conclude that the proportion of defective items in the shipment is larger than 0.1. Thus, Company B can reject the shipment.