Answer to Question #107085 in Math for Moore

Question

Quebec is the only province in Canada where the one official language is French. According to a recent census, 79.7% of Quebec residents identify French as their mother tongue. You select an SRS of 165 Quebec residents. What is the probability that less than 80% of them identify French as their mother tongue? (Identify the mean, standard deviation, justification that the large counts condition is met, z-score, and probability).

Answer 1

Mean: $\mu=p=0.797.$

Standard deviation: $\sigma=\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.797(1-0.797)}{165}}=0.0313.$

Large counts justification:

$np=165*0.797=131.5>10,\;\\n(1-p)=165(1-0.797)=33.5>10.$

Z-score: $z=\frac{\bar x-\mu}{\sigma}=\frac{0.8-0.797}{0.0313}=0.096.$

Probability: $P(X<0.8)=P(Z<0.096)=0.5398.$