Question

Suppose you take a random sample of 30 individuals from a large population. For this sample, the sample mean is 4.2 and sample variance is 49. You wish to estimate the unknown population mean µ.

(a) Calculate a 90% confidence interval for µ.

(b) Calculate a 95% confidence interval for µ.

(c) Based on (a) and (b), comment on what happens to the width of a confidence interval (increase/decrease) when you increase your confidence level.

(d) Suppose your sample size is 100 instead of 30. The sample mean and variance are still 4.2 and 49 respectively. Calculate a new 90% confidence interval for µ.

(e) Based on (a) and (d), comment on what happens to the width of a confidence interval (increase/decrease) when you increase your sample size.

Answer 1

Given that

sample n =30

sample mean μ,=4.2

sample variance σ^2= 49

standard deviation σ = 7

a)

Calculate high end confidence intenvaltotal High End -X tscore s/Nn High End = 4.2 + 1.6991 x 7-30 High End 4.21.6991 x 1.2780193008454 High End 4.22.1714825940664 High End 6.37 Calculate low end confidence interval total Low End X-tscore s/hn Low End = 4.2-1.6991 x 7/:30 Low End = 4.2-1.6991 x 1.2780 193008454 Low End = 4.2-2.171 4825940664 Low End = 2.03 Now we have everything, display our 90% confidence interval 2.03 <H < 6.37

b)

High End a X+tscorea High End 4.2 2.0452 x 7N30 High End 4.2 2.0452 x 1.2780193008454 High End 4.2 2.613805074089 High End 6.81 Calculate low end confidence interval total Low End tscorea * s/In Low End 4.2-2.0452 x730 Low End 4.2-2.0452 x 1.2780 193008454 Low End 4.2 2.613805074089 Low End 1.59 Now we have everything. display our 95% confidence interval 1.59 <p < 6.81

c)  The width increases as the confidence level increase

d)

Calculate high end confidence interval total: High End X+ zscores shn High End 4.21.645 7/N100 High End = 4.2 + 1.645 * 0.7 High End 4.21.1515 1.1515 can be derived on Excel below In Microsoft Excei or Googie Sheets, you write this function as CONFIDENCE(0.1,7,100) High End 5.35 end confidence interval total Low End-X-zscoreo" s/vn Low End = 4.2-1.645 * 7A100 Low End = 4.2-1.645 * 0.7 Low End 4.2-1.1 515 Low End 3.05 Now we have everything, display our 90% confidence interval: 3.05 < <5.35

e)

Increasing the sample size decreases the width of confidence intervals