Questions: Let X represent the full height of a certain species of tree. Assume that X has a normal probability distribution with mean 175.8 ft and standard deviation 37.9 ft. You intend to measure a random sample of n=219 trees. The bell curve below represents the distribution of these sample means. The scale on the horizontal axis is the standard error of the sampling distribution. Complete the indicated boxes, correct to two decimal places.

Let X represent the full height of a certain species of tree. Assume that X has a normal probability distribution with mean 175.8 ft and standard deviation 37.9 ft.

You intend to measure a random sample of n=219 trees. The bell curve below represents the distribution of these sample means. The scale on the horizontal axis is the standard error of the sampling distribution. Complete the indicated boxes, correct to two decimal places.

Transcript text: Let $X$ represent the full height of a certain species of tree. Assume that $X$ has a normal probability distribution with mean 175.8 ft and standard deviation 37.9 ft . You intend to measure a random sample of $n=219$ trees. The bell curve below represents the distibution of these sample means. The scale on the horizontal axis is the standard error of the sampling distribution. Complete the indicated boxes, correct to two decimal places.

Solution

Solution Steps

Step 1: Identify Parameters

The population mean ($\mu$) is 175.8, the population standard deviation ($\sigma$) is 37.9, and the sample size ($n$) is 219.

Step 2: Apply the Central Limit Theorem

Given that the sample size is sufficiently large (usually $n \geq 30$) and the population distribution is normal, the distribution of the sample means will also be normal.

Step 3: Calculate the Standard Error (SEM)

The standard error of the mean ($\sigma_{\bar{X}}$) is calculated as $\frac{\sigma}{\sqrt{n}} = 2.56$.

Step 4: Determine the Sampling Distribution

The sampling distribution of the sample mean will have a mean equal to $\mu = 175.8$ and a standard deviation equal to $\sigma_{\bar{X}} = 2.56$.

Step 5: Find Probabilities

Use the properties of the normal distribution to find probabilities related to the sample means. This involves calculating z-scores and using standard normal distribution tables or software.

Final Answer:

The sampling distribution of the sample mean for a sample size of 219 from a population with mean 175.8 and standard deviation 37.9 is normally distributed with mean $\mu = 175.8$ and standard error $\sigma_{\bar{X}} = 2.56$.

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