A sample of blood is assayed for its alcohol content (BAC). The following BAC values are determined from six trials:

0.081%, 0.078%, 0.079%, 0.076%, 0.082%, 0.077%

Assume the data follows a Gaussian distribution. For 84% of the possible assay data population determine with 93% confidence:
a. The two sided tolerance interval
b. The one sided upper bound tolerance interval
c. The one sided lower bound tolerance interval

To determine the tolerance intervals for the given data, we need to compute the standard deviation (σ) and the mean (μ) of the data set. We can then use these values to calculate the tolerance intervals based on the desired confidence level.

Step 1: Calculate the mean (μ) of the data set.
To calculate the mean, add up all the BAC values and divide the sum by the number of trials (n).

μ = (0.081 + 0.078 + 0.079 + 0.076 + 0.082 + 0.077) / 6 = 0.0795

Step 2: Calculate the standard deviation (σ) of the data set.
The standard deviation can be calculated using the following formula:

σ = √[(Σ(x - μ)²) / (n - 1)]

where:
Σ is the sum of,
x is each individual BAC value, and
μ is the mean value calculated in step 1.

Substituting the given values:

σ = √[((0.081 - 0.0795)² + (0.078 - 0.0795)² + (0.079 - 0.0795)² + (0.076 - 0.0795)² + (0.082 - 0.0795)² + (0.077 - 0.0795)²) / (6 - 1)]

σ ≈ 0.00245

Step 3: Calculate the tolerance intervals.

a. Two-sided tolerance interval:
To determine the two-sided tolerance interval, we need to find the upper and lower bounds within which a certain percentage of the population falls. In this case, we are looking for a 84% coverage with 93% confidence.

Using a statistical table or software, we find that the critical value for a 93% confidence level is approximately 1.81.

The two-sided tolerance interval can be calculated using the following formula:

Lower bound = μ - (Z * σ)
Upper bound = μ + (Z * σ)

where:
Z is the critical value for the given confidence level.

Lower bound = 0.0795 - (1.81 * 0.00245) ≈ 0.0790
Upper bound = 0.0795 + (1.81 * 0.00245) ≈ 0.0800

Therefore, the two-sided tolerance interval is between 0.0790% and 0.0800%.

b. One-sided upper bound tolerance interval:
To determine the one-sided upper bound tolerance interval, we only need to find the upper bound within which a certain percentage of the population falls. In this case, we are looking for a 84% coverage with 93% confidence.

The one-sided upper bound tolerance interval can be calculated using the following formula:

Upper bound = μ + (Z * σ)

where:
Z is the critical value for the given confidence level.

Upper bound = 0.0795 + (1.81 * 0.00245) ≈ 0.0800

Therefore, the one-sided upper bound tolerance interval is above 0.0800%.

c. One-sided lower bound tolerance interval:
The one-sided lower bound tolerance interval can be calculated using the following formula:

Lower bound = μ - (Z * σ)

where:
Z is the critical value for the given confidence level.

Lower bound = 0.0795 - (1.81 * 0.00245) ≈ 0.0790

Therefore, the one-sided lower bound tolerance interval is below 0.0790%.

In summary:
a. The two-sided tolerance interval is between 0.0790% and 0.0800%.
b. The one-sided upper bound tolerance interval is above 0.0800%.
c. The one-sided lower bound tolerance interval is below 0.0790%.