Question

In a market region half of the households is known to use a particular brand of soap. In a household survey, a sample of $10$ house holds are alloted to each investigator and $2048$ investigators are appointed for the survey. The number of investigators likely to report that there are at least $4$ users is

• A. $240$
• B. $352$
• C. $1696$
• D. $120$

Answer 1

The correct option is C $1696$

$E\{X=n\}=\left(\begin{array}{c}10\\ n\end{array}\right)\ast {\frac{1}{2}}^n\ast {\frac{1}{2}}^{10-n}\ast (no.ofinvestigator)E\{X=0\}+E\{X=1\}+E\{X=2\}+E\{X=3\}=\left(\begin{array}{c}10\\ 0\end{array}\right)\ast {\frac{1}{2}}^0\ast {\frac{1}{2}}^{10}\ast 2048+\left(\begin{array}{c}10\\ 1\end{array}\right)\ast {\frac{1}{2}}^1\ast {\frac{1}{2}}^9\ast 2048\left(\begin{array}{c}10\\ 2\end{array}\right)\ast {\frac{1}{2}}^2\ast {\frac{1}{2}}^8\ast 2048+\left(\begin{array}{c}10\\ 3\end{array}\right)\ast {\frac{1}{2}}^3\ast {\frac{1}{2}}^7\ast 2048E\{X=0\}+E\{X=1\}+E\{X=2\}+E\{X=3\}=352Soforatleast4users=1-atmost3=2048-352=1696$