Question

The United States Centers for Disease Control and Prevention (CDC) found that 17.9% of women ages 12 – 59 test seropositive for HPV‑16. Suppose that Tara, an infectious disease specialist, assays blood serum from a random sample of ?=1000 women in the United States aged 12 – 59. Apply the central limit theorem for the distribution of a sample proportion to find the probability that the proportion, ?̂ , of women in Tara's sample who test positive for HPV‑16 is greater than 0.199 . Express the result as a decimal precise to three places. ?(?̂ >0.199)= Apply the central limit theorem for the distribution of a sample proportion to find the probability that the proportion of women in Tara's sample who test positive for HPV‑16 is less than 0.174 . Express the result as a decimal precise to three places. ?(?̂ <0.174)=

Answer 1

Here p = 0.179 and n = 1000

standard error, se = sqrt(p*(1-p)/n)
= sqrt(0.179*(1-0.179)/1000)
= 0.0121

Using central limit theorem,
z = (pcap - p)/se

1)
P(pcap > 0.199)
z = (0.199 - 0.179)/0.0121
z = 1.6529

P(pcap > 0.199)
= P(z > 1.6529)
= 0.0492 (Using right tailed z-table)
= 0.049 (three decimal place)

2)
P(pcap < 0.174)
z = (0.174 - 0.179)/0.0121
z = -0.4132

P(pcap < 0.174)
= P(z < -0.4132)
= 0.3397 (Using left tailed z-table)
= 0.340 (three decimal place)