An interactive poll found that 345345 of 2 comma 3832383 adu

An interactive poll found that 345345 of 2 comma 3832,383 adults aged 18 or older have at least one tattoo. (a) Obtain a point estimate for the proportion of adults who have at least one tattoo. (b) Construct a 9595% confidence interval for the proportion of adults with at least one tattoo. (c) Construct a 9898% confidence interval for the proportion of adults with at least one tattoo. (d) What is the effect of increasing the level of confidence on the width of the interval?

Solution

Given that an interactive poll found that 345345 of 2 comma 3832,383 adults aged 18 or older have at least one tattoo.

Total number of adults aged 18 or older have at least one tattoo (n) = 3832,383

number of adults aged 18 or older have at least one tattoo (x) = 345345

Therefore a point estimate for the population proportion of adults who have at least one tattoo is p^ (sample proportion).

sample proportion (p^) = x / n = 345345 / 3832383 = 0.0901

q^ = 1 - p^ = 1 - 0.0901 = 0.9099

Construct a 95% confidence interval for the proportion of adults with at least one tattoo.

95% confidence interval for the proportion of adults with at least one tattoo is ,

p^ - E < P < p^ + E

where E is margin of error.

E = Zc * sqrt( (p^*q^)/n )

Zc is the critical value for standard normal distribution.

c = confidence level = 95% or 0.95

Zc = 1.96

E = 1.96 * sqrt [ ( 0.0901 * 0.9099) / 3832383 ]

E = 1.96 * sqrt( 2.13948 E-08)

E = 1.96 * 0.00014627

E = 0.000286689

Lower limit = p^ - E = 0.0901 - 0.000286689 = 0.08981

Upper limit = p^ + E = 0.0901 - 0.000286689 = 0.09040

length of CI = upper limit - lower limit = 0.0006

Construct a 98% confidence interval for the proportion of adults with at least one tattoo.

Zc at 98% confidence level is,

Zc = 2.33

E = 2.33 * 0.00014627 = 0.000340809

lower limit = p^ - E = 0.0901 - 0.000340809 = 0.08977

uppelr limit = p^ + E = 0.0901 + 0.000340809 = 0.09045

length of CI = upper limit - lower limit = 0.0007

So we see that as confidence level increases length of confidence in terval increases.