Question

An article in the Australian Journal of Agricultural Research, "Non-Starch Polysaccharides and Broiler Performance on Dlets Containing Soyabean Meal as the Sole Protein Concentrate" (1993, Vol, 44, No. 8, pp. 1483-1499) determined the essential amino acid (Lysine) composition level of soybean meals are as shown below (g/ kg) : Round your answers to 2 decimal places. (a) Construct a 99% two-sided confidence interval for σ 2

≤σ 2
≤ (b) Calculate a 99% lower confidence bound for σ 2
. ≤σ 2
(c) Calculate a 95% lower confidence bound for σ. ≤σ

Answer 1

To calculate confidence intervals for the variance
`($\sigma^2$)` and the standard deviation
`($\sigma$)` of the Lysine composition levels in soybean meals, we'll use the Chi-Square distribution.

Given data:

Column A: 22.2, 24.7, 20.9, 26.0, 27.0

Column B: 24.8, 26.5, 23.8, 25.6, 23.9

Let's first calculate the sample variance
`($s^2$)` of the Lysine composition levels.

Step 1: Calculate the sample variance.

Sample variance formula:

`\[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \]`

Where:

`\( x_i \) =` each value in the sample

`\( \bar{x} \) =` sample mean

`\( n \) =` number of observations

For Column A:

`\( \bar{x}_A = (22.2 + 24.7 + 20.9 + 26.0 + 27.0)/(5) = (120.8)/(5) = 24.16 \)`

`\[ s_A^2 = ((22.2 - 24.16)^2 + (24.7 - 24.16)^2 + (20.9 - 24.16)^2 + (26.0 - 24.16)^2 + (27.0 - 24.16)^2)/(5 - 1) \]`

`\[ s_A^2 = (4.0256 + 0.0289 + 12.0964 + 3.3664 + 8.6436)/(4) \]`

`\[ s_A^2 = (28.16)/(4) = 7.04 \]`

For Column B:

`\( \bar{x}_B = (24.8 + 26.5 + 23.8 + 25.6 + 23.9)/(5) = (124.6)/(5) = 24.92 \)`

`\[ s_B^2 = ((24.8 - 24.92)^2 + (26.5 - 24.92)^2 + (23.8 - 24.92)^2 + (25.6 - 24.92)^2 + (23.9 - 24.92)^2)/(5 - 1) \]`

`\[ s_B^2 = (0.0144 + 2.4784 + 1.3664 + 2.0736 + 1.0564)/(4) \]`

`\[ s_B^2 = (7.9892)/(4) = 1.9973 \]`

Step 2: Calculate the degrees of freedom.

Degrees of freedom
`(\(df\))` for sample variances is
`\(n - 1\)`.

For both Column A and Column B,
`\(df = 5 - 1 = 4\)`.

(a) Construct a 99% two-sided confidence interval for
`\( \sigma^2 \)`.

The formula for the confidence interval for variance using Chi-Square distribution is:

`\[ \left( ((n-1)s^2)/(\chi^2_(\alpha/2)), ((n-1)s^2)/(\chi^2_(1-\alpha/2)) \right) \]`

Where
`\( \chi^2_(\alpha/2) \)` and
`\( \chi^2_(1-\alpha/2) \)` are the Chi-Square critical values at
`\( \alpha/2 \)` and
`\( 1-\alpha/2 \)` percentiles respectively with
`\( df \)` degrees of freedom.

For a 99% confidence interval with
`\( df = 4 \)`, the critical values are
`\( \chi^2_(0.005) \)` and
`\( \chi^2_(0.995) \)`.

We need to find these critical values using a Chi-Square distribution table or statistical software.

For
`\( df = 4 \)`:

`\( \chi^2_(0.005) = 0.484 \) (approx.)`

`\( \chi^2_(0.995) = 13.277 \) (approx.)`

Now plug these values into the formula for both Column A and Column B:

For Column A:

`\( \left( (4 * 7.04)/(13.277), (4 * 7.04)/(0.484) \right) \)`

`\( \left( 2.131, 58.024 \right) \)`

For Column B:

`\( \left( (4 * 1.9973)/(13.277), (4 * 1.9973)/(0.484) \right) \)`

`\( \left( 0.602, 15.448 \right) \)`

(b) Calculate a 99% lower confidence bound for
`\( \sigma^2 \)`.

The lower confidence bound for variance is simply the lower limit of the confidence interval, which is the first value in the interval.

For Column A: Lower bound
`\(= 2.131\)`

For Column B: Lower bound
`\(= 0.602\)`

(c) Calculate a 95% lower confidence bound for
`\( \sigma \)`.

The formula for the lower confidence bound for standard deviation
`(\( \sigma \))` from the variance
`(\( \sigma^2 \))` is:

`\[ \sqrt{\text{Lower bound of } \sigma^2} \]`

For Column A:
`\( √(2.131) \approx 1.46 \)`

For Column B:
`\( √(0.602) \approx 0.775 \)`

These values represent the lower bounds for the standard deviation
`(\( \sigma \))` at 95% confidence level.

An article in the Australion Journal of Agricultural Research, "Non-Starch Polysaccharides and Broiler Performance on Diets Containing Soyabean Meal as the Sole Protein Concentrate
`$(1993, \mathrm{Vol}, 44, \mathrm{Na}, 8, \mathrm{pp} .1483-1499)$` determined the essential amino acid (Lysine) composition level of soybean meals are as shown below (g/kg):

Round your answers to 2 decimal places.

(a) Construct a
`$99 \%$` two-sided confidence interval for
`$\sigma^2$`.

(b) Calculate a
`$99 \%$` lower confidence bound for
`$\sigma^2$`

(c) Calculate a
`$95 \%$` lower confidence bound for
`$\sigma$`.

The table is given below: