Question

Consider a population having a standard deviation equal to 9.94. We wish to estimate the mean of this population.

(a) How large a random sample is needed to construct a 95% confidence interval for the mean of this population with a margin of error equal to 1? (Round your answer to the next whole number.)

The random sample is __________ units.

(b) Suppose that we now take a random sample of the size we have determined in part a. If we obtain a sample mean equal to 345, calculate the 95% confidence interval for the population mean. What is the interval’s margin of error? (Round your answers to the nearest whole number.)

The 95% confidence interval is            [ , ] .

Margin of error = ____________

Answer 1

Solution :

Given that,

standard deviation = = 9.94

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z_/2 = Z_0.025 = 1.96

(a)

margin of error = E = 1

Sample size = n = ((Z_/2 * ) / E)²

= ((1.96 * 9.94) / 1)²

= 379.56 = 380

The random sample is = 380 units .

(b)

= 345

n = 380

Margin of error = E = Z_/2* ( /n)

= 1.96 * (9.94 / 380)

= 1

Margin of error = 1

At 95% confidence interval estimate of the population mean is,

- E < < + E

345 - 1 < < 345 + 1

344 < < 346

(344 , 346)