Question

We want to investigate the proportion of the adult American who state that the amount of tax (federal, state, property, sales, and so on) they pay is too high. In a survey of 1200 adult Americans, 640 state that the amount of tax they pay is too high. To finish the investigation do just one of the following.

(a) Use a 95% confidence interval to determine the proportion of Americans that state the amount of tax they pay is too high and determine if this percentage is more than 50% or not.

(b) Use hypothesis testing at the significance level α = 0.025 to see if the common belief that most Americans state that the amount of tax they pay is too high can be contradicted.

please show your work

Answer 1

a)

p̂ = X / n = 640/1200 = 0.533
p̂ ± Z(α/2) √( (p * q) / n)
0.533 ± Z(0.05/2) √( (0.5333 * 0.4667) / 1200)
Z(α/2) = Z(0.05/2) = 1.96
Lower Limit = 0.533 - Z(0.05) √( (0.5333 * 0.4667) / 1200) = 0.505
upper Limit = 0.533 + Z(0.05) √( (0.5333 * 0.4667) / 1200) = 0.562
95% Confidence interval is ( 0.505 , 0.562 )

Since 0.50 does not contained in confidence interval, and all values in confidence interval

are greater than 0.50, we conclude that percentage is more than 50%.

b)

H0: p <= 0.50

Ha: p > 0.50

Sample proportion = 640 / 1200 = 0.5333

Test statistics

z = ( - p) / sqrt [ p( 1 - p) / n ]

= (0.5333 - 0.50) / sqrt [ 0.5 * 0.5 / 1200 ]

= 2.31

Critical value at 0.025 level = 1.96

Since test statistics > 1.96 , Reject H0.

We conclude that we have sufficient evidence to conclude that most Americans state that the amount of tax

they pay is too high can be contradicted.