OpenIntro_q5.13

David Simbandumwe
October 13, 2021

Question 5.13

5.13 Website registration. A website is trying to increase registration for first-time visitors, exposing 1% of these visitors to a new site design. Of 752 randomly sampled visitors over a month who saw the new design, 64 registered.

a) Check any conditions required for constructing a confidence interval.

b) Compute the standard error.

c) Construct and interpret a 90% confidence interval for the fraction of first-time visitors of the site who would register under the new design (assuming stable behaviors by new visitors over time).

a) confidence interval conditions

Check any conditions required for constructing a confidence interval.

1. Independence: Sampled observations must be independent

random sampling/assignment is used, and
if sampling without replacement, n < 10% of the population.

2. Sample size: There should be at least 10 expected successes and 10 expected failures in the observed sample. {np >= 10} - {n(1-p) >= 10}

n <- 752
x <- 64
(p_hat <- x/n)

[1] 0.08510638

(p_hat * n >= 10)

[1] TRUE

(n * (1 - p_hat) >= 10)

[1] TRUE

b) standard error

Compute the standard error.

\( SE = \sqrt{p(1-p)/n} \)

x <- 64
n <- 752

(p_hat = x/n)

[1] 0.08510638

(se = round(sqrt(p_hat * ( 1 - p_hat) / n),4))

[1] 0.0102

c) 90% confidence interval

Construct and interpret a 90% confidence interval for the fraction of first-time visitors of the site who would register under the new design (assuming stable behaviors by new visitors over time).

p_hat (+/-) z * se

(z <- -qnorm(0.05))

[1] 1.644854

(ci <- c( round(p_hat - z * se,4),round(p_hat + z * se,4)))

[1] 0.0683 0.1019