A manufacturer of t-shirts marks a shirt as “irregular” when it has defects such as crooked seams, stains, rips, or holes. A small number of irregular t-shirts are expected as part of the manufacturing process, but if more than 8% of the t-shirts manufactured at a plant are classified as irregular, the manager has to do an investigation to try to find the source of the increased mistakes in the manufacturing process.In order to test whether his plant is making a higher than expected number of irregular t-shirts, the manager of a plant randomly selects 100 t-shirts and finds that 12 are irregular.He plans to test the hypotheses: H0, P = 0.08, versus Ha, p > 0.08 (where p is the true proportion of irregular t-shirts). What is the test statistic?
Question
A manufacturer of t-shirts marks a shirt as “irregular” when it has defects such as crooked seams, stains, rips, or holes. A small number of irregular t-shirts are expected as part of the manufacturing process, but if more than 8% of the t-shirts manufactured at a plant are classified as irregular, the manager has to do an investigation to try to find the source of the increased mistakes in the manufacturing process.In order to test whether his plant is making a higher than expected number of irregular t-shirts, the manager of a plant randomly selects 100 t-shirts and finds that 12 are irregular.He plans to test the hypotheses: H0, P = 0.08, versus Ha, p > 0.08 (where p is the true proportion of irregular t-shirts). What is the test statistic?
Solution
The test statistic for a hypothesis test for a proportion is a z-score (z). This is calculated using the formula:
z = (p̂ - P0) / sqrt((P0 * (1 - P0)) / n)
where:
- p̂ is the sample proportion
- P0 is the hypothesized population proportion
- n is the sample size
In this case:
- p̂ = 12/100 = 0.12 (the proportion of irregular t-shirts in the sample)
- P0 = 0.08 (the hypothesized population proportion of irregular t-shirts)
- n = 100 (the number of t-shirts in the sample)
Substituting these values into the formula gives:
z = (0.12 - 0.08) / sqrt((0.08 * (1 - 0.08)) / 100) z = 0.04 / sqrt(0.0736) z = 0.04 / 0.27129319932501076 z = 0.14744195615489714
So, the test statistic is approximately 0.147.
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