An article reports that when each football helmet in a random sample of 34 suspension-type helmets was subjected to a certain impact test, 24 showed damage. Let p denote the proportion of all helmets of this type that would show damage tested in the prescribed manner. Required:
a. Calculate a 99% Cl for p.
b. What sample size would be required for the width of a 99% Cl to beat most .10, irrespective of p ?

a

b

Step-by-step explanation:

From the question we are told that

    The sample size is  n =  34

    The  number of damaged helmets is  x  =  24

Now the proportion of damaged helmets is mathematically represented as

substituting values

Given that the confidence level is 99% the level of significance can be evaluated as

     %

Next we obtain the critical value of    from the normal distribution table, the value is  

The reason we are obtaining critical values of   instead of   is because  

represents the area under the normal curve where the confidence level interval (  ) did not cover which include both the left and right tail while  

is just the area of one tail which what we required to calculate the margin of error

   The margin of error is mathematically represented as

substituting values

The 99% confidence interval for p is mathematically represented as

substituting values

The  sample size required for the width of a 99% Cl to beat most 0.10, irrespective of p ? is mathematically represented as  

Here    telling  us that the deviation from the sample proportion is  set to 0.10 irrespective of the value of  

so the  sample size for this condition is

=>