Question

The sugar content of the syrup in canned peaches is normally distributed. A random sample of n = 10 cans yields a sample standard deviation of s = 4.848 milligrams. Construct a 95% two-sided confidence interval for σ. Round your answers to 2 decimal places.

Answer 1

Answer:

95% Confidence interval of the standard deviation: 3.33 ≤ σ ≤ 8.85

Step-by-step explanation:

Sample size = 10

Degrees of freedom = 10-1 = 9

For a 95% confidence interval, we have  α=0.05, which gives 2.5% of the area at each end of the chi-square distribution.

We calculate the chi-square values for a 95% confidence interval

Value of chi-square_0.975 = 2.70

Value of chi-square_0.025 = 19.02

We then evaluate $[tex]\sqrt{ \frac{(n-1)*s^{2} }{X^{2} }}$[/tex] for the two values of chi-square.

Then we have:

$[tex]\sqrt{\frac{(n-1)*s^{2} }{X^{2}}} =\sqrt{\frac{(10-1)*4.848^{2} }{2.70}}=\sqrt{78.34}=8.85\\\\\\sqrt{\frac{(n-1)*s^{2} }{X^{2}}} =\sqrt{\frac{(10-1)*4.848^{2} }{19.02}}=\sqrt{11.12}=3.33$[/tex]

With these results we can express the confidence interval as

3.33 ≤ σ ≤ 8.85