A VP of manufacturing claims that more than 3% of items produced using a certain assembly line are is defective. A random sample of 100 items from the assembly line had 4 defectives. At the 5% level of significance, which of the following would be part of the standardized decision rule?

Question

A VP of manufacturing claims that more than 3% of items produced using a certain assembly line are is defective. A random sample of 100 items from the assembly line had 4 defectives. At the 5% level of significance, which of the following would be part of the standardized decision rule?

a) Conclude Ho if TS is greater than or equal to 1.645

b) Conclude Ho if TS is less than or equal to 1.645

c) Conclude Ho if TS is greater than or equal to 1.960

d) Conclude Ho if TS is less than or equal to 1.960

Answer 1

To answer this question, we need to perform a hypothesis test.

First, let's define the null hypothesis (Ho) and the alternative hypothesis (Ha).

Ho: The proportion of defective items produced using the assembly line is 3% or less.
Ha: The proportion of defective items produced using the assembly line is greater than 3%.

Next, we need to calculate the test statistic (TS). The test statistic for a hypothesis test involving a proportion is given by:

TS = (p̂ - p) / √(p * (1-p) / n)

Where:
p̂ is the sample proportion of defectives (number of defectives / sample size)
p is the hypothesized population proportion (3% in this case)
n is the sample size (100 in this case)

In our case, the sample had 4 defectives out of 100 items, so p̂ = 4/100 = 0.04.

Now, we can calculate the test statistic:

TS = (0.04 - 0.03) / √(0.03 * (1-0.03) / 100) = 0.01 / √(0.0291) ≈ 0.182

Now, we can compare the test statistic to the critical value to make a decision. In this case, we are testing the alternative hypothesis that the proportion is greater than 3%, so we should use a one-tailed test.

The standardized decision rule depends on the desired significance level. The question states that we want to use a 5% level of significance, which means that the critical value corresponds to a 5% right-tail probability.

Looking up the critical value for a 5% right-tail probability in a standard normal distribution table, we find that it is approximately 1.645.

Since the standardized decision rule is asking for the part of the rule that is true, we need to select the option that matches the given critical value.

The correct answer is (a) Conclude Ho if TS is greater than or equal to 1.645.