A random sample of 60 suspension helmets used by motorcycle riders and automobile race-car drivers was subjected to an impact test, and on 16 of these helmets some damage was observed. (a) Find a 95% two-sided confidence interval on the true proportion of helmets of this type that would show damage from this test. Round your answers to 3 decimal places. ≤p≤ (b) Using the point estimate of p obtained from the preliminary sample of 60 helmets, how many helmets must be tested to be 95% confident that the error in estimating the true value of p is less than 0.02? n= (c) How large must the sample be if we wish to be at least 95% confident that the error in estimating p is less than 0.02, regardless of the true value of p?

Answer:a)0.154<p<0.48[b)n≅ 6.57c)n=2401Step-by-step explanation:n = 60Sample proportion = p = 16/60 = 0.266a) Find a 95% two-sided confidence interval on the true proportion of helmets 90 % confidence interval fro population is [tex]p +- z0.05/2 * \sqrt{p(1-p)/n}[/tex][tex]0.154<p<0.48[/tex]hence population proportion lies in the confidence intervalb) helmets must be tested to be 95% confident point of estimates i s- 0.266Margin of error = 0.02[tex]ME=z0.05/2*\sqrt{p(1-p)n}[/tex]after putting values we getn = 6.56n≅ 6.57c)large must the sample beThe sample is calculted as [tex]n=(z0.025/E)^{2} 1/4[/tex][tex]n=(1.96/0.02)^{2} 1/4[/tex]n=2401