Question

A company in the food industry stores a large number of canned goods in a central warehouse. Last year, 3% of the canned goods had damage (for example, ugly worms or dents in the can), and the warehouse manager suspects it could be even worse this year.

To investigate this, we randomly selected 260 of the preserves and of these, 13 have
damage.
(a) Conduct hypothesis testing to test if the warehouse manager's suspicions can be considered where
Acknowledged.
(b) Formulate an interpretation of the P-value for the test in the (a) assignment. (The P value is
probability of ...)
(c) If we did the investigation, how could we do it to get higher?
strength of the test in the (a) assignment? Justify the answer.
(d) Describe what a Type I error and a Type II error would mean in this context.

Answer 1

(a) Here hypothesis are

H0 : p = 0.03

Ha : p > 0.03

sample proportion = p^ = 13/260 = 0.05

standard error of proportion = sep = sqrt(0.03 * 0.97/260) = 0.0106

Test statistic

z = (0.05 - 0.03)/0.0106 = 1.8905

p- value = P(Z > 1.8906) =1 - NORMSDIST(1.8905) = 1 - 0.9707 = 0.0293

so here as we see that p value is less than significance level, we would reject the null hypothesis and conclude that manager's suspicions can be considered where Acknowledged.

(b) Here p - value of 0.0293 means that there is 2.93% probability that we will get a sample results as provided when the true defective percentage is 3%.

(c) To get a higher strength, we will take more sample size alongwith lower significance level.

(d) Here type I error is when we reject the null hypothesis that means we would conclude that mean defective rate is higher than 3% but in actual it is equal to 3%.

Type II error is when we fail to reject the null hypothesis that means we would conclude that mean defective rate is not higher than 3% but in actual it is higher than 3%.