Question

Estimates of the earth's biomass, the total amount of vegetation held by the earth's forests, are important in determining the amount of unabsorbed carbon dioxide that is expected to remain in the earth's atmosphere. \(^{2}\) Suppose a sample of 75 one-square-meter plots, randomly chosen in North America's boreal (northern) forests, produced a mean biomass of 4.2 kilograms per square meter \(\left(\mathrm{kg} / \mathrm{m}^{2}\right)\), with a standard deviation of \(1.5 \mathrm{~kg} / \mathrm{m}^{2}\). Estimate the average biomass for the boreal forests of North America and find the margin of error for your estimate.

Short answer

Answer: The estimated average biomass for the boreal forests of North America is about 4.2 kg/m², with a margin of error of ±0.3451 kg/m² at a 95% confidence level.

Step-by-step solution

Identify the sample mean, sample size, and sample standard deviation

In this problem, we are given the following values: Sample mean (\(\bar{x}\)) = 4.2 kg/m² Sample size (n) = 75 Sample standard deviation (s) = 1.5 kg/m²

Determine the appropriate t-critical value

Since we don't know the population standard deviation, we will use the t-distribution to find the t-critical value for a 95% confidence interval. The degrees of freedom (df) is equal to the sample size minus one: df = n - 1 = 75 - 1 = 74. For a 95% confidence interval and 74 degrees of freedom, the t-critical value (t) can be found using a t-table or calculator, which yields t ≈ 1.993.

Calculate the standard error

The standard error (SE) is equal to the sample standard deviation divided by the square root of the sample size: \(\text{SE} = \frac{s}{\sqrt{n}} = \frac{1.5}{\sqrt{75}} ≈ 0.1732\) kg/m²

Calculate the margin of error

The margin of error (ME) is equal to the product of the t-critical value and the standard error: \(\text{ME} = t \times \text{SE} = 1.993 \times 0.1732 ≈ 0.3451\) kg/m²

Estimate the average biomass and margin of error

The estimate for the average biomass for the boreal forests of North America is equal to the sample mean: Average biomass (μ) ≈ 4.2 kg/m² The margin of error for our estimate is: ± 0.3451 kg/m² Thus, we can conclude that the average biomass for the boreal forests of North America is about 4.2 kg/m², with a margin of error of ±0.3451 kg/m² at a 95% confidence level.

Key concepts

Confidence Interval

When estimating the average biomass of North America's boreal forests from a sample, we express our degree of certainty in the form of a confidence interval. This statistical range offers an assessed probability that the true average biomass falls within that interval. The confidence interval is calculated using the sample mean, the critical value from the t-distribution (or z-distribution if the population standard deviation is known), and the standard error of the mean.

A common confidence level used in practice is 95%, which implies that if we repeated our sampling and interval calculations many times, about 95 out of 100 of those intervals would contain the true mean value. Confidence intervals provide a useful way to convey the range within which the true mean is likely to lie, taking into account sample variability.

Sample Mean

The sample mean, often denoted as \(\bar{x}\), is a key statistic that represents the average value of a sample. In the case of estimating biomass, the sample mean is the total biomass from the sample plots divided by the number of plots (4.2 kg/m²). This value is then used as the point estimate for the true mean biomass of the entire region. It's important to remember that the sample mean is only an estimate, and as such, there is always some uncertainty associated with it, which is why we calculate a margin of error and construct a confidence interval around the sample mean.

T-distribution

The t-distribution, also known as Student's t-distribution, is a probability distribution that accounts for the added uncertainty that comes from estimating the population standard deviation from the sample. Unlike the normal distribution, the t-distribution is slightly wider and has heavier tails, which means it predicts a greater chance for extreme values as the sample size decreases, which compensates for the uncertainty.

As the sample size gets larger, the t-distribution approaches a normal distribution shape. The t-distribution requires the specification of 'degrees of freedom', which for a one-sample t-test, is the sample size minus one (\(n - 1\)). In our exercise with a sample size of 75, the degrees of freedom was 74. The t-distribution is especially important when dealing with small sample sizes or when the population standard deviation is unknown.

Margin of Error

The margin of error reflects the extent of uncertainty around the sample mean. It quantifies the range either side of the sample mean within which we can be a certain percentage sure that the true mean lies. The larger the margin of error, the less precise the estimate.

The margin of error is influenced by the size of the sample, the variability in the data, and the level of confidence desired in the estimate. It's computed by taking the product of the critical value from the t-distribution and the standard error of the sample mean. In our context, a margin of error of ±0.3451 kg/m² suggests that the true mean biomass of the boreal forests is most likely between 3.8549 kg/m² and 4.5451 kg/m², with a 95% level of confidence.