Question

1.Describe the sampling distribution of proportions.

2.What do you understand by the expression "p is an unbiased estimator of p"?

3.What is meant by the phrase "95% confidence interval for the population p"?

4.A consumer watch group interviewed a random sample of 1,000 consumers and asked them if they are optimistic about the economy or not. There were 612 who said yes. Use 95% confidence level to estimate the population proportions.

Answer 1

1.The sampling distribution of a given population is the distribution of frequencies of a range of different outcomes that could possibly occur for a statistic of a population. It describes a range of possible outcomes that of a statistic, such as the mean or mode of some variable, as it truly exists a population.

2. If p has the minimum variance among all the unbiased estimators of p, so It can be said that p is an unbaise estimator of p.

3.A 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population.

4.$n=1000, p=\dfrac{612}{1000}=0.612, q=1-p=1-0.612=0.388$

$\alpha=0.05\\ Z_{\frac{\alpha}{2}}=Z_{0.025}=1.96$

Confidence interval = $p\pm Z_{0.025}\sqrt{\dfrac{pq}{n}}$

= $0.612\pm 1.96\times \sqrt{\dfrac{0.612\times 0.388}{1000}}$

$=0.612\pm 0.0302=(0.58179,0.64220)$