Question

Cancer survival rates indicate the percentage of people who survive a certain type of cancer for a specific amount of time. Cancer statistics often use an overall 5-year survival rate. For instance, the overall 5-year survival rate for bladder cancer is 75%. That means that of all people diagnosed with bladder cancer, 75 out of every 100 were able to survive beyond 5 years.

Suppose that there was a new treatment for bladder cancer and a study was conducted to examine whether it can improve on the current 5-year survival rate. A sample of 125 people diagnosed with bladder cancer were selected to take part in the trial, with a total of 95 still alive five years after their initial diagnosis.

Let p denote the proportion of people with bladder cancer who survived for at least 5 years under the new treatment.

(a) Determine the proportion of people who survived for 5 years after diagnosis.

  = .   (exact value)

(b) In order to calculate a confidence interval for p, it is necessary to be able to approximate the probability distribution of the number of 5-year bladder cancer survivers using the normal distribution. Calculate the following:

  = ;      (exact value)

  = .      (exact value)

Hence, conclude whether the normal approximation is valid.

Is the approximation valid?  (type Yes or No):

(c) Specify the value of k required to calculate the 95% confidence interval for p.

k = .      (to 3 decimal places)

(d) Hence, determine the 95% confidence interval for p.

< p < .       (to 3 decimal places)

Answer 1

(a)

the proportion of people who survived for 5 years after diagnosis = = 95/125 = 0.76

So,

Answer is:

0.76

(b)

np = 125 X 0.76 = 95 10

n (1 - p) = 125 X 0.24 = 30 10

Hence, conclude whether the normal approximation is valid.

Is the approximation valid?

Answer :

Yes

(c)

From Table, critical values of Z = 1.96

So,

k = 1.96

(d)

Confidence Interval:

So,

Answer is:

0.685 < p < 0.835