In the united states, paper currency often comes into contact with cocaine directly, during drug deals or usage, or in counting machines where it wears off from one bill to another. A forensic survey collected fifty $1 bills and measured the cocaine content of the bills. Forty-six of the bills had measurable amounts of cocaine on them. Assume that the sample of bills was a random sample, each bill has a equal chance of being picked, each is picked independently of the others and each has the same chance of being picked. Please answer the following questions.
(a) Suppose that the US Treasury Department claims that 4 out of 5 circulating American 1$ bills contain residues of cocaine. Is there any evidence based on these results that the claim is incorrect? Note: find the p-value using pbinomor dbinom.
(b) For the test in (a), is there evidence of incorrectness at level 0.01? Explain, briefly.
(c) What sort of error could you be making in (a)? Explain briefly, in the context of the problem.
Answer:
Given,
p = 4/5
= 0.8
alpha = 0.05
Null hypothesis Ho : p = 0.8
Alternative hypothesis Ha : p != 0.8
x = 46
sample n = 50
sample proportion p^ = x/n
= 46/50
= 0.92
consider,
test statistic z = (p^ - p)/sqrt(p(1-p)/n)
substitute values
= (0.92 - 0.8)/sqrt(0.8(1-0.8)/50)
z = 2.1213
Corresponding P value = 0.0338966 [since from z table]
= 0.0339
Here p value < alpha, so we reject Ho.
We have sufficient evidence.
b)
Here significance level = 0.01
P value is greater than significance level, so we fail to reject Ho.
There is no sufficient evidence.
c)
To a limited extent, we dismiss the Ho, so we are making type 1 mistake i.e., error
Type mistake happens when we dismiss a genuine Ho.
We are dismissing that 4 out of 5 circling American 1$ bills contain buildups of cocaine while it is valid.
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