Question

Can you taste PTC? PTC is a substance that has a strong bitter taste for some people and is tasteless for others. The ability to taste PTC is inherited. About $75\%$of Italians can taste PTC, for example. You want to estimate the proportion of Americans who have at least one Italian grandparent and who can taste PTC.

a. How large a sample must you test to estimate the proportion of PTC tasters within $0.04$with $90\%$confidence? Answer this question using the $75\%$estimate as the guessed value for $\hat{p}$.

b. Answer the question in part (a) again, but this time use the conservative guess$\hat{p}=0.5$. By how much do the two sample sizes differ?

Short answer

a. $n=318$

b. Sample size increases by$105$.

Step-by-step solution

Given Information

It is given that $E=0.04,\hat{p}=0.50$

$1-\alpha =0.90$, using table$z_{a/2}=1.645$

Calculating Sample Size

It is calculated as: $\hat{p}\text{known:}n=\frac{{\left[z_{a/2}\right]}^2\hat{p}\hat{q}}{E^2}$

$=\frac{{\left[z_{w/2}\right]}^2\hat{p}(1-\hat{p})}{E^2}$

and $\hat{p}\text{unknown:}n=\frac{{\left[z_{a/2}\right]}^{2}0.25}{E^2}$

As $\hat{p}$is known, sample size

$n=\frac{{\left[z_{a/2}\right]}^2\hat{p}(1-\hat{p})}{E^2}$

$=\frac{1.{645}^2\times 0.75(1-0.75)}{0.{04}^2}$

$n=318$

Increase in Sample Size

Also, $n=\frac{{\left[z_{a/2}\right]}^2\hat{p}(1-\hat{p})}{E^2}$

$=\frac{1.{645}^2\times 0.50(1-0.50)}{0.{04}^2}$

$=423$

Difference between two sizes, $423-318$

$=105$

Sample size increases by$105$