Question

(1 point) Working backwards, Part I. A 90% confidence interval for a population mean is (77, 85). The population distribution is approximately normal and the population standard deviation is unknown. This confidence interval is based on a simple random sample of 27 observations. Calculate the sample mean, the margin of error, and the sample standard deviation. Use the t distribution in any calculations. Round non-integer results to 4 decimal places.

Sample mean =

Margin of error =

Sample standard deviation =

Answer 1

Solution:

A 90% confidence interval for a population mean is (77, 85)

Upper limit = 85

Lower limit = 77

Since population SD is unknown , this interval is constructed using t distribution.

n = 27

c = 0.90

= 1- c = 1- 0.90 = 0.10

  /2 = 0.10 2 = 0.05

Also, d.f = n - 1 = 27 - 1 = 26  

    =    =  _0.05,26 =  1.706 ..use t table

1) Sample mean = (Upper limit + Lower limit)/2

= (85 + 77)/2

= 81

2)Margin of error = (Upper limit - Lower limit)/2

= (85 - 77)/2

= 4

3)Find Sample standard deviation s

We know , margin of error =   /2,d.f. * ( / n)

Bur from part 2, margin of error = 4

4 = /2,d.f. * ( / n)

4 = 1.706 * ( / 27)

s = (4 * 27) /1.706

= 12.1832