3. In studying the efficiency of the federal court system, we look at the random variable, X, the amount of time elapsing between the filing of a charge and the trial date. A random sample of n = 86 cases yields a sample mean of 11.4 months, with sample variance s2x = 5.8 months.
(a) Find a 90% confidence interval on the average time between the filing of charges and the date of trial.
(b) Find a 95% confidence interval on ¦Ìx.
(c) What is the probability that a case would require at least one year between the filing of charges and the date of trial?
See previous posts.
(a) To find a 90% confidence interval on the average time between the filing of charges and the date of trial, we can use the formula for a confidence interval:
CI = X̄ ± Z * (s / √n)
where X̄ is the sample mean, Z is the z-score for the desired confidence level, s is the sample standard deviation, and n is the sample size.
Given that X̄ = 11.4 months, s = √s2x = √5.8 months, and n = 86, we need to find the z-score for a 90% confidence level.
Using a standard normal distribution table or a calculator, we find that the z-score for a 90% confidence level (with a two-tailed test) is approximately 1.645.
Plugging in the values, the confidence interval is:
CI = 11.4 ± 1.645 * (√5.8 / √86)
Simplifying the expression, we get:
CI = 11.4 ± 1.645 * 0.678
Thus, the 90% confidence interval on the average time between the filing of charges and the date of trial is approximately:
CI = (10.076, 12.724) months
(b) To find a 95% confidence interval on σx, the standard deviation of X, we can use the formula for a confidence interval:
CI = (n - 1) * s² / χ²(L/2, n - 1) , (n - 1) * s² / χ²(1 - L/2, n - 1)
where s² is the sample variance, n is the sample size, and χ²(L/2, n - 1) and χ²(1 - L/2, n - 1) are the chi-square values for the desired confidence level (L) and degrees of freedom (n - 1). Here L = 0.95.
Given that n = 86 and s² = 5.8 months, we need to find the chi-square values for the 95% confidence level.
Using a chi-square distribution table or a calculator, we find that the chi-square value for the 95% confidence level and 85 degrees of freedom is approximately 107.85 (χ²(0.025, 85)) and 60.19 (χ²(0.975, 85)).
Plugging in the values, the confidence interval is:
CI = 85 * 5.8 / 107.85 , 85 * 5.8 / 60.19
Simplifying the expression, we get:
CI ≈ (4.567, 8.731) months
Thus, the 95% confidence interval on σx is approximately:
CI = (4.567, 8.731) months
(c) To find the probability that a case would require at least one year (12 months) between the filing of charges and the date of trial, we need to calculate the cumulative probability of X being greater than or equal to 12 months.
First, we need to standardize the variable X by using the following formula:
Z = (X - μ) / σ
where X is the specific value (12 months in this case), μ is the population mean, and σ is the population standard deviation. Since we only have the sample mean (X̄) and variance (s²x), we can make an approximation by assuming that X follows approximately a normal distribution.
The standardized value is:
Z = (12 - X̄) / s
where X̄ = 11.4 months and s = √s²x = √5.8 months.
Plugging in the values, we get:
Z = (12 - 11.4) / √5.8
Simplifying the expression, we get:
Z ≈ 0.376
Now, we can use a standard normal distribution table or a calculator to find the cumulative probability of Z being greater than or equal to 0.376.
Using the table or calculator, we find that the cumulative probability is approximately 0.6461.
Thus, the probability that a case would require at least one year between the filing of charges and the date of trial is approximately 0.6461 or 64.61%.