Question

botanists are concerned that rising temperatures are impeding the growth of trees in taylor national forest. to investigate, they measured the heights of a random sample of 100 mature birch trees to the nearest tenth of a meter. the botanists found that the average height of these trees was 12.8 meters, and the standard deviation was 6.4 meters. find a 99% confidence interval for the mean height of mature birch trees in taylor national forest. round your answers to the nearest tenth. \boxed{} < \mu < \boxed{}

Explanation:

Step1: Identify the formula for confidence interval

For a confidence interval for the population mean when the population standard deviation is unknown (we use the sample standard deviation \( s \)) and the sample size \( n \) is large (\( n \geq 30 \)), we use the formula:

$$ \bar{x} \pm z_{\alpha/2} \frac{s}{\sqrt{n}} $$

where \( \bar{x} \) is the sample mean, \( z_{\alpha/2} \) is the z - score corresponding to the level of confidence, \( s \) is the sample standard deviation, and \( n \) is the sample size.

Step2: Determine the values of the variables

We are given that:

• The sample mean \( \bar{x}=12.8 \) meters.
• The sample standard deviation \( s = 6.4 \) meters.
• The sample size \( n = 100 \).
• For a 99% confidence interval, the significance level \( \alpha=1 - 0.99 = 0.01 \), so \( \alpha/2=0.005 \). The \( z \) - score \( z_{0.005} \) (the z - score such that the area to the right of it is 0.005) is approximately 2.576 (we can find this value from the standard normal distribution table).

Step3: Calculate the margin of error \( E \)

First, calculate \( \frac{s}{\sqrt{n}} \):

$$ \frac{s}{\sqrt{n}}=\frac{6.4}{\sqrt{100}}=\frac{6.4}{10} = 0.64 $$

Then, calculate the margin of error \( E=z_{\alpha/2}\times\frac{s}{\sqrt{n}} \):

$$ E = 2.576\times0.64=1.64864 $$

Step4: Calculate the confidence interval

The lower limit of the confidence interval is \( \bar{x}-E \):

$$ 12.8 - 1.64864=11.15136\approx11.2 $$

The upper limit of the confidence interval is \( \bar{x}+E \):

$$ 12.8+1.64864 = 14.44864\approx14.4 $$

Answer:

\( 11.2<\mu<14.4 \)