95% one-sided upper confidence interval for the population variance is (0, 0.0235].
This means that we are 95% confident that the true population variance of fish fry weight is less than or equal to 0.0235.
Since the calculated upper bound of the confidence interval (0.0235) is less than the manager's desired maximum variance of 0.05, the sample supports the manager's belief that the variance of fish fry weight is indeed within acceptable limits.
Formula for one-sided upper confidence interval for variance:
The formula for a one-sided upper confidence interval for variance (σ²) with confidence level 1 - α (e.g., 95%) is:
σ² ≤ s² + [(n - 1) / χ²(α, n - 1)] * s²
where:
σ² is the population variance
s² is the sample variance (0.014 in this case)
n is the sample size (10)
χ²(α, n - 1) is the upper χ² critical value for α and n - 1 degrees of freedom. For a 95% confidence level and 9 degrees of freedom (n - 1), the critical value is approximately 14.684.
Calculating the confidence interval:
Plugging in the values:
σ² ≤ 0.014 + [(10 - 1) / 14.684] * 0.014 ≈ 0.014 + 0.0095 ≈ 0.0235
Therefore, the 95% one-sided upper confidence interval for the population variance is (0, 0.0235].
Interpretation:
This means that we are 95% confident that the true population variance of fish fry weight is less than or equal to 0.0235.
Since the calculated upper bound of the confidence interval (0.0235) is less than the manager's desired maximum variance of 0.05, the sample supports the manager's belief that the variance of fish fry weight is indeed within acceptable limits.