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Chapter 8: Problem 8

A simple random sample of size \(n=40\) is obtained from a population with \(\mu=50\) and \(\sigma=4 .\) Does the population need to be normally distributed for the sampling distribution of \(\bar{x}\) to be approximately normally distributed? Why? What is the sampling distribution of \(\bar{x} ?\)

Short Answer

Expert verified
No, the population doesn't need to be normally distributed since \(n=40\). The sampling distribution of \(\bar{x}\) is approximately normal with mean 50 and standard error \(0.632\).

Step by step solution

01

Analyze the Population Distribution Requirement

The Central Limit Theory (CLT) states that the sampling distribution of the sample mean \(\bar{x}\) becomes approximately normally distributed, regardless of the population distribution, if the sample size is large enough, typically \(n \geq 30\). Since \(n=40\), which is greater than 30, the population does not need to be normally distributed for the sampling distribution of the sample mean to be approximately normal.
02

Determine the Sampling Distribution

The sampling distribution of \(\bar{x}\) will follow a normal distribution as per the CLT. The mean of the sampling distribution, \(\mu_\bar{x}\), is equal to the population mean \(\mu=50\). The standard deviation of the sampling distribution, known as the standard error \(\sigma_\bar{x}\), is calculated with the formula \(\sigma_\bar{x} = \frac{\sigma}{\sqrt{n}}\).
03

Calculate the Standard Error

Using the provided values, the standard error of the mean is \(\sigma_\bar{x} = \frac{4}{\sqrt{40}} \approx 0.632\).
04

Summarize the Sampling Distribution

The sampling distribution of \(\bar{x}\) is approximately normal with mean \(\mu_\bar{x} = 50\) and standard error \(\sigma_\bar{x} \approx 0.632\).

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Key Concepts

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

sampling distribution
In statistics, a sampling distribution is the probability distribution of a given statistic based on a random sample. This distribution is crucial because it forms the foundation for statistical inference. It's essential to understand that a sampling distribution creates a collection of sample means (or other statistics) from all possible samples of the same size from a population.
Imagine repeatedly drawing samples from a population and calculating the mean for each sample. If you plot these sample means, you get the sampling distribution of the sample mean. The shape and spread of this distribution depend on the size of the sample and the population parameters.
sample mean
The sample mean, denoted as \(\bar{x}\), is simply the average of the values in a sample. It serves as an estimate of the population mean \(\mu\).
In the given example, let's assume we have a sample size of 40 drawn at random from a population. If we sum all the values in this sample and divide by 40, we get the sample mean.
  • The sample mean is used because it provides a good point estimate of the population mean.
  • It's critical in inference because it forms the basis of the sampling distribution of the mean.
standard error
Standard error measures the variability of the sample mean around the population mean. It is denoted as \(\sigma_{\bar{x}}\). Calculation of the standard error involves the population standard deviation (\(\sigma\)) and the sample size (\(n\)):
\[\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\]
The smaller the standard error, the closer the sample means are to the population mean. In our exercise, using \(\sigma = 4\) and \(n = 40\), the standard error is calculated as:
\[\frac{4}{\sqrt{40}} \approx 0.632\]
normal distribution
Normal distribution is fundamental in statistics. It describes how values are distributed in a dataset. This bell-shaped curve shows most data points clustering around the mean.
The Central Limit Theorem (CLT) is essential here. It states that, given a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the population's distribution.
  • This is why, in our exercise, the sample size (\(n=40\)) ensures the sampling distribution of the mean is approximately normal.
  • This normality allows us to make various inferences about the population mean.

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Most popular questions from this chapter

Determine \(\mu_{\bar{x}}\) and \(\sigma_{\bar{x}}\) from the given parameters of the population and the sample size. \(\mu=52, \sigma=10, n=21\)

The reading speed of second grade students is approximately normal, with a mean of 90 words per minute (wpm) and a standard deviation of 10 wpm. (a) What is the probability a randomly selected student will read more than 95 words per minute? (b) What is the probability that a random sample of 12 second grade students results in a mean reading rate of more than 95 words per minute? (c) What is the probability that a random sample of 24 second grade students results in a mean reading rate of more than 95 words per minute? (d) What effect does increasing the sample size have on the probability? Provide an explanation for this result. (e) A teacher instituted a new reading program at school. After 10 weeks in the program, it was found that the mean reading speed of a random sample of 20 second grade students was 92.8 wpm. What might you conclude based on this result? (f) There is a \(5 \%\) chance that the mean reading speed of a random sample of 20 second grade students will exceed what value?

A simple random sample of size \(n=36\) is obtained from a population with \(\mu=64\) and \(\sigma=18\). (a) Describe the sampling distribution of \(\bar{x}\). (b) What is \(P(\bar{x}<62.6) ?\) (c) What is \(P(\bar{x} \geq 68.7) ?\) (d) What is \(P(59.8<\bar{x}<65.9) ?\)

Describe the sampling distribution of \(\hat{p}\). Assume that the size of the population is 25,000 for each problem. $$ n=1000, p=0.103 $$

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