91Ó°ÊÓ

A random sample of \(n=12\) four-year-old red pine trees was selected, and the diameter (in inches) of each tree's main stem was measured. The resulting observations are as follows: \(\begin{array}{llllllll}11.3 & 10.7 & 12.4 & 15.2 & 10.1 & 12.1 & 16.2 & 10.5\end{array}\) \(\begin{array}{llll}11.4 & 11.0 & 10.7 & 12.0\end{array}\) a. Compute a point estimate of \(\sigma\), the population standard deviation of main stem diameter. What statistic did you use to obtain your estimate? b. Making no assumptions about the shape of the population distribution of diameters, give a point estimate for the population median diameter. What statistic did you use to obtain the estimate? c. Suppose that the population distribution of diameter is symmetric but with heavier tails than the normal distribution. Give a point estimate of the population mean diameter based on a statistic that gives some protection against the presence of outliers in the sample. What statistic did you use? d. Suppose that the diameter distribution is normal. Then the 90 th percentile of the diameter distribution is \(\mu+1.28 \sigma\) (so \(90 \%\) of all trees have diameters less than this value). Compute a point estimate

Step by step solution

01

Calculate Sample Standard Deviation

Start by computing the sample mean \( \bar{X} \). Next, subtract the sample mean from each data point and square the result. Add up these squared values, divide by \( n-1 \) where n is the sample size (12 in this case), and take the square root. The result is the sample standard deviation \( s \), which serves as an estimate for \( \sigma \).

02

Calculate Sample Median

Sort the data points in ascending order. For a dataset with an even number of observations, the median is the average of the two middle values. The median in this case is our point estimate for the population median.

03

Calculate Sample Trimmed Mean

First, remove the highest and lowest values from our dataset. This is done as these values may represent outliers, which can heavily influence our sample mean. After removing these values, calculate the mean of the remaining data points. This is known as the trimmed mean and is a point estimate for the population mean, assuming a symmetric distribution with heavy tails.

04

Calculate 90th Percentile

Assuming normality, the 90th percentile can be calculated using the expression \( \mu + 1.28\sigma \). Substitute the sample mean for \( \mu \) and the sample standard deviation for \( \sigma \). The result serves as a point estimate for the 90th percentile of the population.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Standard Deviation

The sample standard deviation is a measure that shows how much the data in a sample typically deviates from the mean. Essentially, it's a snapshot of variability or spread within the sample. To calculate it, follow these steps:

• Find the mean (average) of your data set.
• Subtract the sample mean from each data point, then square the result.
• Sum these squared differences.
• Divide this sum by the number of data points minus one (\( n-1 \)).
• Take the square root of this quotient to get the sample standard deviation (\( s \)).

This method gives you a point estimate for the population standard deviation (\( \sigma \)).
When you have a small sample size like 12 trees, the sample standard deviation is often slightly different from the true population standard deviation. It's important in statistics as it gives us an idea about the spread of the sample data.

Population Median

The population median is a statistic that represents the middle value in a set of data, ensuring half of the values lie below and half above it. To estimate the population median from a sample, sort your data in ascending order.
Since the number of observations here is even (12 trees), the median is found by taking the average of the 6th and 7th values in this ordered list.
This simple calculation provides a robust point estimate of the population median, particularly useful when the data is skewed or has outliers, which can often distort other measures like the mean.

Trimmed Mean

A trimmed mean is a type of average that removes the extreme values or outliers, thereby giving a more "typical" mean of the data. It's particularly beneficial when the data is symmetric but has heavy tails, where outliers are possible.

• Sort the data points in ascending order.
• Remove the highest and lowest values, which are potential outliers.
• Calculate the mean of the remaining data points.

By using the trimmed mean, you get a point estimate of the population mean that is less affected by the outliers, making it a reliable measure for distributions that are not perfectly normal. This process helps ensure the result is reflective of the general data trend without distortion from extreme values.

90th Percentile

The 90th percentile is a statistic that shows the value below which 90% of the data falls. It gives insights into the upper tail of your data distribution. For normally distributed data, such as assumed here, it can be calculated using the formula:\[\text{90th percentile} = \mu + 1.28\sigma\]Where \( \mu \) is the sample mean and \( \sigma \) is the sample standard deviation. Substituting the sample mean and standard deviation into this formula provides a point estimate of the 90th percentile.
This percentile is especially useful in understanding the extremes of your dataset, such as the largest tree diameters in a forest, but only if the data is assumed to follow a normal distribution.