Question

Following are the numbers of grams of sugar per 100 grams of apple in a random sample of 6 Red Delicious apples. Assume the population is normally distributed. Construct a 95% confidence interval for the standard deviation of the number of grams of sugar. Round the answers to two decimal places.

12.3 12.5 13.3 13.5 11.8 10.6

Answer 1

Confidence interval for population standard deviation is given as below:

Sqrt[(n – 1)*S2 / χ2 α/2, n – 1 ] < σ < sqrt[(n – 1)*S2 / χ2 1 - α/2, n – 1 ]

We are given

Confidence level = 95%

Sample size = n = 6

Degrees of freedom = n – 1 = 5

Sample standard deviation = S = 1.05956

χ2 α/2, n – 1 = 12.8325

χ2 1 - α/2, n – 1 = 0.8312

(By using chi square table)

Sqrt[(n – 1)*S2 / χ2 α/2, n – 1 ] < σ < sqrt[(n – 1)*S2 / χ2 1 - α/2, n – 1 ]

Sqrt[(6 – 1)* 1.05956^2 / 12.8325] < σ < sqrt[(6 – 1)* 1.05956^2 / 0.8312]

0.6614 < σ < 2.5987

Lower limit = 0.66

Upper limit = 2.60