For each of these problems, conduct a significance test. Remember there are four
ID: 3217085 • Letter: F
Question
For each of these problems, conduct a significance test. Remember there are four steps after confirming the conditions are met for inference:
1) state the hypotheses
2) calculate test statistic
3) find P-value. Remember that we are using = 0.05 as our guideline for statistical significance. If P-value 0.05, reject Ho. If P-value > 0.05, do not reject Ho.
4) state the conclusion in plain English in the context of the problem, not just “reject Ho” or “do not reject Ho.” Look at the statements for Problem 1 and use them as a template for your conclusions. If you decide to reject Ho give the P-value.
Problem 1) The reputations (and hence sales) of many businesses can be severely damaged by shipments of manufactured items that contain a large percentage of defectives. For example, a manufacturer of alkaline batteries may want to be reasonably certain that fewer than 6% of its batteries are defective. Suppose a random sample of 300 batteries are selected from a very large shipment; each is tested and 5 defective batteries are found. (The manufacturer is analyzing the proportion of DEFECTIVE batteries.)
a) Consider just one experimental unit – that is, one battery. What is the response variable for that one battery? Categorical or quantitative?
b) Verify the three conditions for using the central limit theorem for inference on .
c) Conduct a significance test to decide if there is sufficient evidence for the manufacturer to conclude that the fraction defective in the entire shipment is less than 6 percent. Be sure to state your conclusion in plain English in the context of the problem. Choose one of the statements for your conclusion, depending on your P-value: We have evidence to show that the fraction defective in the entire shipment is less than 6 percent (P-value = _______) We do not have evidence to show that the fraction defective in the entire shipment is less than 6 percent.
Explanation / Answer
a) categorical as a battery is either defective or not. No numerical characteristics involved
b) Randomization: This condition is assumed as the problem states that a random sample was selected
10% Rule: The problem statement mentions "sample containing 300 batteries out a large shipment" and therefore is safe to assume 300 is less than 10% of the population
np0=300*.06=18, nq0= 300*.94=282; As np0 and nq0 are greater than 10, we can approximate the sampling distribution using normal approximation
c) p0 = 0.06; p = 5/300 = 1/60, n = 300
H0 : p=p0, Ha : p<p0
Left-tailed, directional hypothesis test
The test statistic formula is:
z = (1/60 - 0.06)/sqrt((0.06*0.94)/300) = -3.1604
p value = 0.0008
We have evidence to show that the fraction defective in the entire shipment is less than 6 percent (P-value = 0.0008)
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