H2 Maths Stats qn

quailmaster SG

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

wee_ws SG

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

wee_ws SG

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

quailmaster SG

for q1 i tried a normal approx X~N(128, 25.6) and got an ans of 0.943

quailmaster SG

oops forgot to apply continuity correction :)

quailmaster SG

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 ?

eagle SG

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.

quailmaster SG

in this case how to apply continuity correction? 8.08 becomes 8.5?

eagle SG

Oops... I see wrong... You didn't use Normal Appromixation, hence no necessity to use continuity correction.

eagle SG

Oops... I see wrong... You didn't use Normal Appromixation, hence no necessity to use continuity correction.

eagle SG

I just calculated. Seems like your answer is correct.

quailmaster SG

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

wee_ws SG

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

eagle SG

Hi Mr Wee

For Q2, it's sufficient to just work with just binomial without using normal approximation right?

wee_ws SG

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

quailmaster SG

thanks for all your help :)