A random sample of 10 observations has been drawn from a normal population, and the sample variance is found to be s² = 19.5. Determine the 98% confidence interval for ó².

Find the 98% confidence interval using a z-table. Find the formula for the confidence interval using standard deviation. Remember that standard deviation is the square root of the variance. Substitute what you know into the formula from the information given to determine the confidence interval.

you are constructing a 95% confidence interval using the information: n = 50, xbar = 54.3, s = 2.1 and E = 0.65. What is the value of the middle of the interval?

To determine the 98% confidence interval for σ² (population variance), we need to use the chi-square distribution.

Step 1: Determine the degrees of freedom.
For the sample variance, we use (n-1) degrees of freedom, where n is the sample size. In this case, n = 10, so the degrees of freedom is 10 - 1 = 9.

Step 2: Find the critical values.
Since the confidence level is 98%, the alpha level (α) is 1 - 0.98 = 0.02. We divide this by 2 to get the two-tailed critical values. So, α/2 = 0.02/2 = 0.01.

To find the critical values from the chi-square distribution table, we need the degrees of freedom and the probability of the left-tail area. We can use the complement of α/2 for this, which is 1 - 0.01 = 0.99.

Using the chi-square distribution table with 9 degrees of freedom and a left tail area of 0.99, we find the critical values to be approximately 2.70 and 19.02.

Step 3: Calculate the confidence interval.
The confidence interval formula for σ² is given by:
[(n-1)*s² / χ²(α/2, n-1), (n-1)*s² / χ²(1-α/2, n-1)]

Plugging in the values, we get:
[(10-1)*19.5 / 19.02, (10-1)*19.5 / 2.70]

Simplifying this, we have:
[8.5, 67.49]

Therefore, the 98% confidence interval for σ² is (8.5, 67.49).

To determine the 98% confidence interval for the population variance (σ²), we can use the Chi-Square distribution. The formula to calculate the confidence interval is as follows:

Lower Bound = [(n - 1) * s²] / χ²(α/2, n - 1)
Upper Bound = [(n - 1) * s²] / χ²(1 - α/2, n - 1)

Where:
- n is the sample size (10 in this case),
- s² is the sample variance (19.5 in this case),
- α is the significance level (1 - confidence level) which is 0.02 for a 98% confidence level,
- χ²(α/2, n-1) is the critical value from the Chi-Square distribution for the lower bound, and
- χ²(1 - α/2, n-1) is the critical value from the Chi-Square distribution for the upper bound.

The critical values can be obtained from a Chi-Square distribution table or calculated using statistical software.

For a 98% confidence level, α/2 is 0.02/2 = 0.01.
So, we need to find the critical value for 0.01 in the Chi-Square distribution with (n - 1) degrees of freedom, which is 9 in this case.

Using a Chi-Square distribution table or statistical software, we find that the critical values are approximately 2.700 for the lower bound and 21.7 for the upper bound.

Substituting the values into the formula:

Lower Bound = [(10 - 1) * 19.5] / 2.700 ≈ 65.00
Upper Bound = [(10 - 1) * 19.5] / 21.7 ≈ 14.29

Therefore, the 98% confidence interval for the population variance (ó²) is approximately (14.29, 65.00).