DM Co. sells computer parts by mail. The company claims that at least 90% of all orders are delivered on time. A sample of 150 orders showed that 129 were mailed on time. Is the company’s claim true? Test using the 2.5% level of significance. Use the 5 step procedure.
To test the company's claim using the 2.5% level of significance, we can follow these steps:
Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha).
- H0: The proportion of orders delivered on time is equal to 90%.
- Ha: The proportion of orders delivered on time is less than 90%.
Step 2: Determine the test statistic.
- In this case, we can use the Z-test statistic for proportions.
Step 3: Set the level of significance (α).
- The question states that we should use the 2.5% level of significance, so α = 0.025.
Step 4: Collect the data and calculate the test statistic.
- From the sample of 150 orders, we know that 129 were mailed on time.
- We can calculate the test statistic using the formula:
z = (p̂ - p) / √((p * (1 - p)) / n)
where p̂ is the sample proportion, p is the hypothesized proportion, and n is the sample size.
- In this case, p̂ = 129/150 = 0.86 (proportion of orders delivered on time in the sample)
and p = 0.90 (proportion claimed by the company).
- Substituting the values into the formula, we get:
z = (0.86 - 0.90) / √((0.90 * (1 - 0.90)) / 150)
z = -0.04 / √((0.09) / 150)
z ≈ -2.14
Step 5: Make a decision and interpret the result.
- We compare the test statistic with the critical value(s) from the standard normal distribution table for a one-tailed test at α = 0.025.
- The critical value for α = 0.025 is approximately -1.96 (since we are testing for less than).
- Since the test statistic (-2.14) is more extreme (smaller) than the critical value (-1.96), we reject the null hypothesis.
- This means that there is sufficient evidence to conclude that the proportion of orders delivered on time is less than 90% as claimed by the company.
Therefore, based on the given information and using the 2.5% level of significance, we can reject the company's claim.