Biomass, the total amount of vegetation held by the earth's...

Biomass, the total amount of vegetation held by the earth's forests, is 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.

Key Concepts

Standard Error

The standard error quantifies the variability of a sample statistic (like the sample mean) around the true population parameter. It is calculated by dividing the sample standard deviation by the square root of the sample size, reflecting how sampling randomness affects the estimate.

Point Estimation

Point estimation involves using a single value, derived from sample data, as the best guess of an unknown population parameter. For instance, using the sample mean to estimate the population mean is a fundamental example of point estimation.

Sampling Distribution

The sampling distribution describes the probability distribution of a sample statistic, such as the sample mean, over many samples from the same population. It is central to inferential statistics because it provides information about the variability and the expected behavior of the estimator.

Margin of Error

The margin of error represents the range within which the true population parameter is expected to lie with a certain level of confidence. It is computed by multiplying the standard error by an appropriate critical value (often from the t-distribution when the population standard deviation is unknown), thereby quantifying the uncertainty in the estimate.

t-Distribution

The t-distribution is used in estimating population parameters when the sample size is relatively small or when the population standard deviation is unknown. It has heavier tails compared to the normal distribution, which provides a more accurate estimation of the critical values needed for constructing confidence intervals with the appropriate level of uncertainty.

Transcript

00:02 All right, with this problem, we know that the basal area measurements are normally distributed.

00:11 So we're going to model with that normal curve.

00:15 And we are going to have a forester sampling nine trees.

00:23 And we know that the standard deviation is approximately four square inches.

00:32 And we want to determine the probability.

00:36 That the sample mean is within two square inches of the population mean.

00:47 So we can display that as an absolute value inequality.

00:55 So basically what we're saying here is the population means in the center, and we want x bar to be within two in either direction.

01:07 So x bar can come all the way out to here or can go all the way up to here is what we're trying to do.

01:13 But we can express that as an absolute value inequality.

01:18 So that's what we're trying to determine.

01:22 So the first thing we're going to do is we're going to change that absolute value inequality into a compound inequality.

01:29 So we're going to say the probability that negative 2 is less than or equal to x, bar minus mu, which is in turn less than or equal to a positive two.

01:43 So what that's doing is it's giving us that wiggle room.

01:46 We can go back or we can go forward.

01:51 Now, if you recall what the z score formula looks like for sampling distributions, it would be x bar minus mu divided by sigma over the square root of n.

02:06 So if you look at our compound inequality and you look at the numerator of the z score formula, they're the same.

02:14 So what we want to do is we want to get our compound inequality to look like the z score.

02:20 So if we can place a sigma divided by square root of n underneath, then we've got our z score.

02:29 But you can't just do that.

02:31 You just can't put it under one part, but what we can do is divide all the parts.

02:36 Of the compound inequality by that expression.

02:44 So we'll have negative 2 over sigma divide by square root of n is less than or equal to z, which is less than or equal to 2 divided by sigma over the square root of n.

02:59 Now we know that in this whole problem, our sigma is going to be 4 and our sample size is going to be 9.

03:08 So we can replace both the sigma and the n with their appropriate values.

03:21 So now we are looking for the probability that z is between negative 2 divided by 4 over the square root of 9 and positive 2 divided by 4 over the square root of 9.

03:36 We can simplify the square root of 9 turns into a 3...

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