Question

An inspector inspects a shipment of medications to determine the efficacy in terms of the proportion p in the shipment that failed to retain full potency after 60 days of production. Unless there is clear evidence that this proportion is less than 0.05, she will reject the shipment. To reach a decision, she will test the following hypotheses using the large-sample test for a population proportion:
H0 : p = 0.05, Ha : p < 0.05
To do so, she selects an SRS of 200 pills. Suppose that eight of the pills have failed to retain their full potency.

The p-value of the test is?

Answer 1

Huge.

Answer 2

these are the options

Answer 3

. 0.2352.
B. 0.2582.
C. 0.4704.
D. 0.5164.

Answer 4

This is a binomial distribution test with mean np and variance npq. n=200 p=.05 and q=1-p. Standard Deviation =sqrt(variance).

Answer 5

so it would be d?

Answer 6

So mean = 200*5% = 10 variance = 95%*10=9.5 and Standard Deviation is sqrt(9.5) = 3.08. Her observation was 8 (2 below the mean of 10) which is 2/3.08 of a standard deviation.

Answer 7

im confused.

Answer 8

so the answer is 2/3.08=.65?

Answer 9

Whatever the z-score of -0.649 standard deviations is is your p value.

Answer 10

thank you!!! will post two more questions.

Answer 11

What did you get?

Answer 12

.382

Answer 13

That is too high.

Answer 14

i did invnorm(.649)

Answer 15

i dont know

Answer 16

I don't know what program that is but in Excel, =NORM.S.DIST(-0.649,1) = 0.2582

Answer 17

thank you so much! thats all i have for now

Answer 18

Yeah, invnorm is for when you know p and need to find the number of standard deviations, this is the opposite.

Answer 19

you really know a lot of about statistics! i have my final tomorrow.

Answer 20

Good luck

Answer 21

ahhhh...im nervous

Answer 22

Do you feel like you understand what I am doing with the np, npq stuff?

Answer 23

honestly, not really

Answer 24

So a binomial distribution occurs when you have a collection of items/observations/whatever where each thing is either yes or no. In this case each pill inspected was either good or bad.

Answer 25

n is the number of items you sample and p is the chance that any one item is bad.

Answer 26

oh, i see. its one way or another.

Answer 27

So this could work for shooting free throws in basketball, for example.

Answer 28

Or manufacturing iPods or whatever.

Answer 29

okay

Answer 30

On average the number of bad pills we find is going to be n*p.

Answer 31

so its the proportion?

Answer 32

If p is 5% and we test 200 pills then we expect that .05 * 200 = 10 pills will be bad, usually.

Answer 33

oh i see

Answer 34

But the binomial distribution can be approximated by a normal distribution if n and p are large enough (which in this case is an okay assumption). And the variance of a Binomial distribution = n*p*(1-p)

Answer 35

that makes sense

Answer 36

what is n* and p*?

Answer 37

* is just a multiplication sign.

Answer 38

n is still the number of tests and p is the expected failure rate. 1-p is the expected success rate.

Answer 39

oh right....that makes sense

Answer 40

are you a teacher?

Answer 41

So in this example the variance of the expected distribution is np(1-p) = 200*(0.05)*(0.95)

Answer 42

No, I'm an Actuary.

Answer 43

that explains it well!

Answer 44

And you understand the relationship between variance and standard deviation, right?