Answer to Question #227981 in Statistics and Probability for Donata

Question

Explain the reasons for sampling using a practical example. (10)

b) Discuss the difference between point estimation and confidence interval estimation

using a practical example. (10)

c) Suppose a census was conducted to determine the impact of COVID-19 on the

operations of local vendors and it found that the mean and variance of the daily

income of all local vendors is N$ 600 and 156250 (N$ squared), respectively. Assume

a sample of 100 local vendors is selected, what is the probability that their sample

mean daily income is within N$ 100 of the population mean daily income.(5)

d) Suppose that the population variance in c) is underestimated, what effect will this

have on the probability calculated in c)? Explain. (5)

Answer 1

(a) 

Sampling is needed, especially when it is near impossible to conduct a census survey. For example, if the researcher wants to estimate the mean income of industrial workers in the USA during the lockdown, it would be nearly impossible to survey all the industry workers, so there is a need to sample survey here. Moreover, if the financial resources are limited, then also census is not a choice. 

(b) 

The difference between point estimation and the confidence interval estimation is that the point estimation gives a value and the interval gives a range of values. For example, the mean height of class 12 students in a school is 5.5 ± 0.4. The value5.5 gives the point estimate, and the interval estimate is (5.1, 5.9). 

(c)

$P(600 -100 < X < 600+100) = P(500 < X < 700)\\ =P (\frac{500 — 600}{\sqrt{\frac{156250}{100}}} <Z< \frac{700 -600}{\sqrt{\frac{156250}{100}}})\\ = P(-2.53 <Z < 2.53) \\ = 2P(Z <2.53)-1 \\ = 0.988594$  

(d) If the variance is underestimated,then for true variance, the z-valuewould increase and hence would cover larger area than part (c), so probability would be more in value in this case.