Answer to Question #215960 in Statistics and Probability for jon

Question

The number of carbohydrates found in a random sample of fast-food entrees is listed below. Is there sufficient evidence to conclude that the variance differs from 100? use the 0.05 level of significance.

53 46 39 39 30

47 38 73 43 41

Answer 1

x= number of carbohydrates

$n=10 \\ \bar{x}= \frac{53+46+...+43+41}{10}=44.9 \\ S^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2 \\ S^2 = \frac{1}{10-1} \times ((53-44.9)^2 + (46-44.9)^2+...+(43-44.9)^2 + (41-44.9)^2) = 135.53$

We have to test:

$H_0: \sigma^2=100 \\ H_1: \sigma^2 ≠ 100 \\ α=0.05$

Test-statistic:

$χ^2 = \frac{(n-1)S^2}{\sigma^2} \\ = \frac{9 \times 135.43}{100} = 12.188$

Critical value $χ^2_{n-1, α/2}=19.023$

$χ^2$ < Critical value

Accept H₀.

There is not sufficient evidence to conclude that the variance differs from 100.