A sample of 400 observations will be taken from an infinite population. The population proportion equals 0.8. The probability that the sample proportion will be greater than 0.75 is
A) 0.8764.
B) 0.1236.
C) 0.9938.
D) 0.0062.

Question

Quizplus · Accepted Answer

The probability that the sample proportion will be greater than 0.75 can be found using the normal approximation to the binomial distribution, given the large sample size (n = 400). The mean of the sampling distribution of the proportion ($\bar{p}$) is the population proportion (p = 0.8), and the standard deviation ($\sigma_{\bar{p}}$) is $\sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.8(0.2)}{400}}$.To find the probability that the sample proportion is greater than 0.75, we calculate the z-score for 0.75: $z = \frac{0.75 - 0.8}{\sqrt{\frac{0.8(0.2)}{400}}}$. Plugging the values into the formula gives a z-score, which we then look up in the standard normal distribution table or use a calculator to find the probability. The calculation will show that the probability is very high, indicating that it is very likely the sample proportion will be greater than 0.75, which corresponds to option C, 0.9938. This is because the sample mean is very close to the population proportion, and given the large sample size, the sampling distribution will be tightly clustered around the population proportion, making values less than 0.75 (more than 0.05 away from the mean) relatively unlikely.

A Sample of 400 Observations Will Be Taken from an Infinite