Question

The main stem growth measured for a sample of seventeen 4 -year-old red pine trees produced a mean and standard deviation equal to 11.3 and 3.4 inches, respectively. Find a \(90 \%\) confidence interval for the mean growth of a population of 4 -year-old red pine trees subjected to similar environmental conditions.

Short answer

Answer: The 90% confidence interval for the mean growth of 4-year-old red pine trees subjected to similar environmental conditions is between 9.858 inches and 12.742 inches.

Step-by-step solution

Calculate the standard error

The first step is to calculate the standard error of the sample mean, which can be found using the formula: $$SE(\bar{X}) = \frac{s}{\sqrt{n}}$$ where \(s\) is the sample standard deviation and \(n\) is the sample size. In our case, \(s = 3.4\) inches and \(n = 17\) trees. Plugging these values into the formula, we get: $$SE(\bar{X}) = \frac{3.4}{\sqrt{17}} = \frac{3.4}{4.12} = 0.826$$

Find the t-value and degrees of freedom

Since we're dealing with a t-distribution, we need to find the t-value corresponding to the 90% confidence level, which can be obtained from the t-table or using a t-distribution calculator. Also, we need to determine the degrees of freedom (df), which can be calculated as follows: $$df = n - 1$$ In this case, the degrees of freedom are: $$df = 17 - 1 = 16$$ Now, we can refer to the t-distribution table for a 90% confidence level and 16 degrees of freedom, which will give us a t-value of approximately 1.746.

Calculate the margin of error

With the standard error and t-value, we can now calculate the margin of error, which can be found using the formula: $$ME = t \cdot SE(\bar{X})$$ Substituting our values, we get: $$ME = 1.746 \cdot 0.826 = 1.442$$

Determine the confidence interval

Now that we have the margin of error, we can calculate the 90% confidence interval for the population mean by adding and subtracting the margin of error from the sample mean: $$CI = (\bar{X} - ME , \bar{X} + ME)$$ $$CI = (11.3 - 1.442 , 11.3 + 1.442) = (9.858, 12.742)$$ So, the 90% confidence interval for the mean growth of 4-year-old red pine trees subjected to similar environmental conditions is between 9.858 inches and 12.742 inches.

Key concepts

Standard Error

The standard error is a crucial concept in statistics. It represents the variability of a sample mean if you were to take multiple samples from a population. In simpler terms, it helps us understand how much the sample mean would differ if we did the experiment over and over again.
To calculate the standard error for our problem, we use the formula \[ SE(\bar{X}) = \frac{s}{\sqrt{n}} \] where

• \(s\) is the sample standard deviation, which was given as 3.4 inches.
• \(n\) is the sample size, which is 17 trees in this case.

Plugging in these numbers, the calculation looks like this: \[ SE(\bar{X}) = \frac{3.4}{\sqrt{17}} = 0.826 \]This number means that the sample mean of tree growth might vary by about 0.826 inches if we were to conduct multiple similar experiments.

T-Distribution

The t-distribution is a type of probability distribution that looks similar to a normal distribution but has fatter tails. This means that it is more spread out compared to a normal distribution. It is particularly useful when dealing with smaller sample sizes and unknown population standard deviations.
We use the t-distribution instead of the normal distribution because it provides a more accurate reflection of what's happening in our sample, given those circumstances.
To find the t-value, which is crucial for calculating our confidence interval, we need to know the degrees of freedom (df). In statistics, degrees of freedom help to determine the shape of the t-distribution. They are calculated as follows:\[ df = n - 1 \] where

• \(n\) is the sample size.
• In our exercise, this gives us \(df = 17 - 1 = 16\).

For a 90% confidence interval and 16 degrees of freedom, we can look up the t-value using statistical tables or calculators, which gives us approximately 1.746.

Margin of Error

The margin of error reflects the range within which we can expect the true population mean to lie, given our sample data. It gives us a sense of how accurate our estimates of the population mean are based on the sample mean we've observed.
To find the margin of error (ME), we use the formula: \[ ME = t \cdot SE(\bar{X}) \] where

• \(t\) is the t-value, which is 1.746 in our case.
• \(SE(\bar{X})\) is the standard error, which we've calculated as 0.826.

Plugging these values into our formula gives us: \[ ME = 1.746 \times 0.826 = 1.442 \]This margin of error means that our confidence interval extends 1.442 inches in both directions from the sample mean. Thus, using this margin, we construct the confidence interval, ensuring it covers where the true mean of the population likely falls.