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 = 4.798 milligrams. Construct a 95% two-sided confidence interval for σ. Round your answers to 2 decimal places.

Answer 1

The 95% confidence interval is given by:

3.30<σ<8.76

Explanation:

1) Data given and notation

s=4.798 represent the sample standard deviation

`\bar x` represent the sample mean

n=10 the sample size

Confidence=95% or 0.95

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population mean or variance lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

The Chi Square distribution is the distribution of the sum of squared standard normal deviates .

2) Calculating the confidence interval

The confidence interval for the population variance is given by the following formula:

`((n-1)s^2)/(\chi^2_(\alpha/2)) \leq \sigma^2 \leq ((n-1)s^2)/(\chi^2_(1-\alpha/2))`

The next step would be calculate the critical values. First we need to calculate the degrees of freedom given by:

`df=n-1=10-1=9`

Since the Confidence is 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a tabel to find the critical values.

The excel commands would be: "=CHISQ.INV(0.025,9)" "=CHISQ.INV(0.975,9)". so for this case the critical values are:

`\chi^2_(\alpha/2)=19.022`

`\chi^2_(1- \alpha/2)=2.700`

And replacing into the formula for the interval we got:

`((9)(4.798)^2)/(19.022) \leq \sigma ((9)(4.798)^2)/(2.700)`

`10.892 \leq \sigma^2 \leq 76.736`

Now we just take square root on both sides of the interval and we got:

`3.30 \leq \sigma \leq 8.76`

So the best option would be:

3.30<σ<8.76