Q1) A dancer performs a certain routine faultlessly 80% of the time. Using a suitable approximation, find the probability that in 160 such occassions, she will perform the routine faultlessly on 75% of the number of such occassions
My working: Y ~ B(160, 0.8)
0.75 * 160 = 120 P( Y > 120) = 1- P(Y<= 120) = 0.928
The ans given by the book is 0.931
Q2) It is claimed that 90% of the children who were given flu jabs do not catch flu in the next 6 mths . A pediatrician gave a flu jab to 150 children in Oct. Find the least integer r suuch that the probability of less than r children catching flu within the next 6 mths is more than 3%
My working
Using the STAT ..Edit function on GC L1: 1 to 10 L2: binomcdf(150, 0.1, L1)
when L1 is 8 L2 is 0.03074
therefore r -1 = 8, r= 9
The ans given by book is 10
Thanks for ur help in advance
Hi,
Q1 requires you to use an approximation. Your working did not show that, so that affected your answer and gave a value different from the book's.
Consider an suitable approximation for Q2 too.
Thanks!
Cheers,
Wen Shih
Hi,
Recall the conditions for various approximations:
Type 1: B(n, p) becomes N(np, np(1 - p)),
provided that n is large, np > 5 and n(1 - p) > 5.
Continuity correction is necessary.
Type 2: B(n, p) becomes Po(np),
provided that n is large and np < 5.
Continuity correction is not necessary.
Type 3: Po(lambda) becomes N(lambda, lambda),
provided that lambda > 10.
Continuity correction is necessary.
Thanks!
Cheers,
Wen Shih
for q1 i tried a normal approx X~N(128, 25.6) and got an ans of 0.943
oops forgot to apply continuity correction :)
for Q2 also use normal approximation ? but i not sure how to apply continuity correction ..
i used InvNorm(0.03, 15, square root 13.5) .. to get 8.09...
need to apply continuity correction for this ?
Hi
if you are using normal approximation to a discrete variable, you need continuity correction.
In question 2, you are using what Mr. Wee has said
Type 1: B(n, p) becomes N(np, np(1 - p)),
provided that n is large, np > 5 and n(1 -
p) > 5.
Continuity correction is necessary.
in this case how to apply continuity correction? 8.08 becomes 8.5?
Oops... I see wrong... You didn't use Normal Appromixation, hence no necessity to use continuity correction.
Oops... I see wrong... You didn't use Normal Appromixation, hence no necessity to use continuity correction.
I just calculated. Seems like your answer is correct.
i approximated the binomial to a normal distribution ..then after that used the invNorm function.. i dunno whether if the approximation i applied in the beginning will affect the final ans
Hi,
Let's look at Q2 again.
Let X represent the number of children, out of 150, catching flu within the next 6 months.
X ~ B(150, 0.1)
which can be approximated by X ~ N(15, 13.5).
Now we want to find least r such that P(X < r) > 0.03.
P(X < r) = P(X <= r - 1) = P(X < r - 1 + 0.5), by continuity correction.
So P(X < r - 0.5) > 0.03.
Since we know the answer. We verify as shown below.
When r = 10, P(X < 9.5) = 0.0672 > 0.03
When r = 9, P(X < 8.5) = 0.0384 > 0.03
When r = 8, P(X < 7.5) = 0.0206 < 0.03
So least r is 9, since beyond that value, we have a probability greater than 0.03.
Thanks!
Cheers,
Wen Shih
Hi Mr Wee
For Q2, it's sufficient to just work with just binomial without using normal approximation right?
Hi,
The method without approximation will give the same answer,
since P(X < 9) = P(X <= 8) = 0.0307.
In general, we use the number of marks as a guide. If not many marks are assigned, we just use the actual probability distribution.
Thanks!
Cheers,
Wen Shih
thanks for all your help :)