Question

2. In a survey of 2096 US adults, 1740 think football teams of all levels should require players who suffer a head injury to take a set amount of time off from playing to recover.

a. Construct a 90% confidence interval for the population proportion. What is the meaning of the confidence interval?

b. Find the minimum sample size needed to estimate the population proportion at the 90% confidence level, using a prior study that found that 70% of U.S. adults think football teams should require players who suffer a head injury to take a set amount of time off from playing to recover. Your estimate must be accurate within 3% of the population proportion.

Answer 1

Part a)

p̂ = X / n = 1740/2096 = 0.8302
p̂ ± Z(α/2) √( (p * q) / n)
0.8302 ± Z(0.1/2) √( (0.8302 * 0.1698) / 2096)
Z(α/2) = Z(0.1/2) = 1.645
Lower Limit = 0.8302 - Z(0.1) √( (0.8302 * 0.1698) / 2096) = 0.8167
upper Limit = 0.8302 + Z(0.1) √( (0.8302 * 0.1698) / 2096) = 0.8436
90% Confidence interval is ( 0.8167 , 0.8436 )
( 0.8167 < P < 0.8436 )

Part b)

p̂ = 0.7
q̂ = 1 - p̂ = 0.3
Critical value Z(α/2) = Z(0.1/2) = 1.6449
n = ( Z(α/2)2 * p̂ * q̂ )/e2
n = ( Z(0.1)2 * 0.7 * 0.3)/ 0.032
n = 632