Question

In a series of 100 tosses of a token, the proportion of heads was found to be 0.58. However, the margin of error for the estimate on the proportion of heads in all tosses was too big. Suppose you want an estimate that is in error by no more than 0.04 at the 95% confidence level. (a) What is the minimum number of tosses required to obtain this type of accuracy? Use the prior sample proportion in your calculation. You should toss the token at least times. (b) What is the minimum number of tosses required to obtain this type of accuracy when you assume no prior knowledge of the sample proportion? You should toss the token at least times.

Answer 1

Solution:

Given:

p =  the proportion of heads = 0.58

E = Margin of Error = 0.04

c = confidence level = 95%

Part a) We have to find the minimum number of tosses required to obtain this type of accuracy using the prior sample proportion in your calculation.

Formula:

We need to find z_c value for c=95% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : Z_c = 1.96

Thus

Part b) We have to find the minimum number of tosses required to obtain this type of accuracy when no prior knowledge of the sample proportion.

If  prior knowledge of the sample proportion is not known,we use p = 0.5

thus we get: