Question

City Planners wish to estimate the mean lifetime of the most commonly planted trees in urban setting. A sample of 16 recently felled trees yielded mean age 32.7 years with standard deviation 3.1 years. Assuming the lifetime of all such trees are normally distributed, construct 98 % confidence intervals for the mean lifetime of all such trees.

Answer 1

The 98% confidence interval for the mean lifetime of the most commonly planted trees in an urban setting, based on a sample mean of 32.7 years and a standard deviation of 3.1 years from 16 samples, is approximately (30.684, 34.716) years.

Step-by-step explanation:

To estimate the mean lifetime of the most commonly planted trees in an urban setting, we use the sample data provided: a mean (sample mean) of 32.7 years, a standard deviation of 3.1 years, and a sample size of 16. We are seeking to construct a 98% confidence interval for the mean lifetime of these trees.

Since the sample size is less than 30 and the population standard deviation is not known, we should use the t-distribution to construct the confidence interval. The degrees of freedom (df) for the t-distribution will be n - 1, where n is the sample size. So df = 16 - 1 = 15.

To find the t-score that corresponds to a 98% confidence level, we consult a t-distribution table, use software, or a calculator, which gives us a t-score around 2.602 for df = 15.

The formula for the confidence interval is:

Confidence interval = ± ( t-score × (s / √n) )

Where:

• s is the sample standard deviation
• n is the sample size

Plugging the values we have:

Confidence interval = ± ( 2.602 × (3.1 / √16) )

Confidence interval = ± ( 2.602 × 0.775) ≈ ± 2.016

Adding and subtracting this interval from the sample mean:

Lower Limit = 32.7 - 2.016 = 30.684 years

Upper Limit = 32.7 + 2.016 = 34.716 years

Therefore, the 98% confidence interval for the mean lifetime of all such trees is (30.684, 34.716) years.