Question

1. Assume that the number of days needed to hatch an egg of a certain type of a rare lizard is distributed normally. Using an incubator, we were able to hatch 13 eggs from different nests. We have a sample mean of 18.97 weeks with a sample standard variance of 10 weeks. What is the 90% confidence interval for the population variance?

Answer 1

Solution :  

Given that,

c = 90%

s² = 10

n = 13

d.f. = n - 1 = 13 -1 = 12

At 90% confidence level ,

$url=http://latex.codecogs.com/gif.latex?%5Calpha&t=1596236557&ymreqid=69d777f6-ff7f-1f97-1c27-8e013201ba00&sig=vzuedYr1btKGmk1FVD9Jow--~D$ = 1 - 90% = 1 - 0.90 = 0.10

$url=http://latex.codecogs.com/gif.latex?%5Calpha&t=1596236557&ymreqid=69d777f6-ff7f-1f97-1c27-8e013201ba00&sig=vzuedYr1btKGmk1FVD9Jow--~D$ / 2 = 0.05 and 1 - ($url=http://latex.codecogs.com/gif.latex?%5Calpha&t=1596236557&ymreqid=69d777f6-ff7f-1f97-1c27-8e013201ba00&sig=vzuedYr1btKGmk1FVD9Jow--~D$ / 2) = 0.95

Now , using chi square table ,

$url=https://latex.codecogs.com/gif.latex?%5Cchi%20%5E%7B2%7D&t=1596236557&ymreqid=69d777f6-ff7f-1f97-1c27-8e013201ba00&sig=0ZWTMbinnUhpe6sSE7JGzQ--~D$_{$url=http://latex.codecogs.com/gif.latex?%5Calpha&t=1596236557&ymreqid=69d777f6-ff7f-1f97-1c27-8e013201ba00&sig=vzuedYr1btKGmk1FVD9Jow--~D$ /2,df} = $url=https://latex.codecogs.com/gif.latex?%5Cchi%20%5E%7B2%7D&t=1596236557&ymreqid=69d777f6-ff7f-1f97-1c27-8e013201ba00&sig=0ZWTMbinnUhpe6sSE7JGzQ--~D$ _0.05,12 = 21.026

$url=https://latex.codecogs.com/gif.latex?%5Cchi%20%5E%7B2%7D&t=1596236557&ymreqid=69d777f6-ff7f-1f97-1c27-8e013201ba00&sig=0ZWTMbinnUhpe6sSE7JGzQ--~D$_{1-$url=http://latex.codecogs.com/gif.latex?%5Calpha&t=1596236557&ymreqid=69d777f6-ff7f-1f97-1c27-8e013201ba00&sig=vzuedYr1btKGmk1FVD9Jow--~D$ /2,df} = $url=https://latex.codecogs.com/gif.latex?%5Cchi%20%5E%7B2%7D&t=1596236557&ymreqid=69d777f6-ff7f-1f97-1c27-8e013201ba00&sig=0ZWTMbinnUhpe6sSE7JGzQ--~D$ _0.95,12 = 5.226

The 90% confidence interval for $\sigma$ ² is,

(n-1)s Xa/2,df < 2 (n-1)s Xi-a/2),df 2

(13 - 1) *10 / 21.026 < $\sigma$ ² < (13 -1 ) *10 /5.226

5.707 < $\sigma$ ² < 22.962

The 90% confidence interval for $\sigma$ ² is  (5.707 , 22.962)