Question

The set in stat3_prob2 presents data collected during a solar energy project at Georgia Tech. (USE R-studio to work on this!)

Set :

y=c(271.8, 264, 238.8, 230.7, 251.6, 257.9, 263.9, 266.5, 229.1, 239.3, 258, 257.6, 267.3, 267, 259.6, 240.4, 227.2, 196, 278.7, 272.3, 267.4, 254.5, 224.7, 181.5, 227.5, 253.6, 263, 265.8, 263.8)

x=c(16.66, 16.46, 17.66, 17.5, 16.4, 16.28, 16.06, 15.93, 16.6, 16.41, 16.17, 15.92, 16.04, 16.19, 16.62, 17.37, 18.12, 18.53, 15.54, 15.7, 16.45, 17.62, 18.12, 19.05, 16.51, 16.02, 15.89, 15.83, 16.71)

(a) Give a point estimate for σ2 and calculate a 95% confidence interval.

Answer 1

1) Point Estimate for ²

By using method of maximum likelihood estimator he point estimate for ² is S².

Where S² is variance of sample

R code :

y=c(271.8,264,238.8,230.7,251.6,257.9,263.9,266.5,229.1,239.3,258,257.6,267.3,267,259.6,240.4,227.2,196,278.7,272.3,267.4,254.5,224.7,181.5,227.5,253.6,263,265.8,263.8)

var(y)

524.3546

x=c(16.66,16.46,17.66,17.5,16.4,16.28,16.06,15.93,16.6,16.41,16.17,15.92,16.04,16.19,16.62,17.37,18.12,18.53,15.54,15.7,16.45,17.62,18.12,19.05,16.51,16.02,15.89,15.83,16.71)

var(x)

0.8247956

The point estimate for ²

for y    = 524.3546

for x = 0.8247956

2) 95% Confidence interval for ²

The 95 % confidence interval for ² is ((n-1)S² /b, (n-1)S² /a)

i) For dataset y

y=c(271.8,264,238.8,230.7,251.6,257.9,263.9,266.5,229.1,239.3,258,257.6,267.3,267,259.6,240.4,227.2,196,278.7,272.3,267.4,254.5,224.7,181.5,227.5,253.6,263,265.8,263.8)

n=25

mean(y)
249.6379
S=(sum(y*y)/24)-(mean^2)
S
73327.69

a=qchisq(0.975,24,lower.tail=FALSE)

a
12.40115

b= qchisq(0.025,24,lower.tail=FALSE)

b
39.36408

((n-1)*S²) /a
141911.4

((n-1*)S²) /b
44707.37

For data y, the confidence interval for ² is (44707.37,141911.4)

ii) For dataset x,

x=c(16.66,16.46,17.66,17.5,16.4,16.28,16.06,15.93,16.6,16.41,16.17,15.92,16.04,16.19,16.62,17.37,18.12,18.53,15.54,15.7,16.45,17.62,18.12,19.05,16.51,16.02,15.89,15.83,16.71)

n=25

mean(x)
[1] 16.70207
S=sum(x*x)/24
S
338.0378
a=qchisq(0.975,24,lower.tail=FALSE)
a
12.40115
b=qchisq(0.025,24,lower.tail=FALSE)
b
39.36408
((n-1)*S)/a
654.2061
((n-1)*S)/b
206.0993

For data y, the confidence interval for ² is (206.0993,654.2061)