Consider the data from an unknown distribution:
40.68, 32.15, 36.91, 29.00, 35.58, 41.55, 33.83, 29.07, 41.00, 37.68, 41.23, 37.54, 40.33, 36.82, 35.18, 31.67, 38.55, 38.34, 35.86, 40.45, 37.28, 36.65, 32.12, 32.96, 37.65, 39.44, 46.18, 24.21, 42.35, 33.05, 30.90, 42.12, 34.38, 37.01, 30.29, 31.12, 31.39, 34.30, 38.65, 42.79
Compute:
a. The two sided tolerance interval
b. What is the level of confidence for this interval if it covers 67% of the data population?
c. Approximately how big a sample size would you need if you wanted to produce an interval that covers 90% of the data's population with 95% confidence? Round your answer to the nearest integer.
To compute the two-sided tolerance interval, we need to follow these steps:
Step 1: Determine the sample mean and standard deviation.
The sample mean, denoted by X̄, is the average of all the data points.
The sample standard deviation, denoted by s, measures the spread of the data points around the mean.
For the given dataset:
Sample Mean (X̄) = (Sum of all data points) / (Number of data points)
= (40.68 + 32.15 + 36.91 + ... + 42.79) / 41
≈ 36.54243902 (round to at least 8 decimal places)
Sample Standard Deviation (s) = sqrt((Sum of (data point - mean)^2) / (Number of data points - 1))
= sqrt(( (40.68 - 36.54243902)^2 + (32.15 - 36.54243902)^2 + ... + (42.79 - 36.54243902)^2 ) / (41 - 1))
≈ 4.31936258 (round to at least 8 decimal places)
Step 2: Determine the critical values for the tolerance interval.
The critical values depend on the desired level of confidence and the number of data points. In a two-sided tolerance interval, we need to take into account the percentage of data points to be covered on both sides.
a. To compute the two-sided tolerance interval, we need to determine the lower and upper critical values.
We can use the qtolerance function in statistical software (e.g., R) or lookup tables for the appropriate critical values.
For example, at a 67% coverage level, the critical values can be found using a statistical software or a lookup table for the t-distribution with (n-1) degrees of freedom, where n is the sample size.
b. To find the level of confidence for the interval that covers 67% of the data population, we need to determine the corresponding critical values. These critical values can be found using a statistical software or lookup table.
c. To determine the sample size required to produce an interval that covers 90% of the data population with 95% confidence, we need to determine the critical values based on the given level of confidence (95%) and coverage level (90%). We can then use a statistical software or a lookup table to find the sample size required.
It's important to note that the exact calculations for the critical values and sample size may vary depending on the specific statistical method or software used.