Question

Use the following to answer questions 19-21 Breast cancer is one of the leading causes of death in women prior to age 50. A study of the effectiveness of regular breast self-examination (BSE) reported in the American Journal of Public Health found that 23 of 53 women who discovered a tumor through BSE had a tumor in an early stage. Of 178 women who discovered a tumor by accident, 48 had a tumor in an early stage. Let p1 and p2 denote the proportion of women who detected early stage tumor through BSE and by accident, and let pi and p denote the corresponding estimates. 19. The SE of pi-D2 is A. 0.07465 B. 0.07358

Answer 1

19)

Option A is correct

Explanation:

The standard error is obtained using the formula,

$SE=\sqrt{\frac{\widehat{P}_{1}\left ( 1- \widehat{P}_{1}\right )}{n_1}+\frac{\widehat{P}_{2}\left ( 1- \widehat{P}_{2}\right )}{n_{2}}}$

Where $\widehat{P}_{1}=\frac{23}{53}\ \ and\ \ \widehat{P}_{2}=\frac{48}{178}$

$SE=\sqrt{\frac{0.434\left ( 1- 0.434\right )}{53}+\frac{0.27\left ( 1- 0.27\right )}{178}}\approx 0.075$

20)

Option (C) is correct

Explanation:

The 95% confidence interval is obtained using the formula,

$95 \%\ confidence\ interval=\left ( \widehat{P}_{1}-\widehat{P}_{2} \right )\pm z^{*}\times SE$

The z-value for 95% confidence interval = 1.96 and SE = 0.075 (with reference to solution 19)

$95 \%\ confidence\ interval=\left ( \frac{23}{53}-\frac{48}{178} \right )\pm 1.96\times 0.075$

$95 \%\ confidence\ interval=0.1642\pm0.147$

21)

Option (D) is correct

Explanation:

The z test is used to compare two proportion. The z-value is obtained as,

$z=\frac{\left (\widehat{P}_{1}-\widehat{P}_{2} \right )-0}{SE}=\frac{0.1642}{0.075}\approx 2.2$

The P-value can be obtained using the standard normal distribution table,

P-value = 0.0113

0.01 < P-value < 0.025