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Chapter 8: Problem 24

Answer the following questions in one or two wellconstructed sentences. a. What happens to the standard error of the mean if the sample size is increased? b. What happens to the distribution of the sample means if the sample size in increased? c. When using the distribution of sample means to estimate the population mean, what is the benefit of using larger sample sizes?

Short Answer

Expert verified
Increasing sample size reduces the standard error, results in a more normal sampling distribution, and improves the accuracy of population mean estimates.

Step by step solution

01

Understanding Standard Error and Sample Size

The standard error of the mean, often denoted as \( SE = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population standard deviation and \( n \) is the sample size, represents the variability of sample means around the population mean.
02

Impact of Increased Sample Size on Standard Error

When the sample size \( n \) is increased, the denominator \( \sqrt{n} \) becomes larger, thus decreasing the standard error \( SE \). This means the sample means are less variable and closer to the population mean.
03

Understanding Distribution of Sample Means

The distribution of the sample means is often referred to as the sampling distribution of the mean. According to the Central Limit Theorem, as the sample size increases, this distribution becomes more normal.
04

Impact of Increased Sample Size on Sampling Distribution

With a larger sample size, the distribution of sample means becomes more narrowly centered around the population mean due to the reduced standard error, leading to a more normal distribution even if the original population is not normal.
05

Benefits of Large Sample Sizes in Estimating Population Mean

Using larger sample sizes allows for a more accurate and reliable estimate of the population mean, as the sampling distribution is less spread out and more closely approximates the actual population mean with less standard error.

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Key Concepts

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

Standard Error
The standard error is a measure of how spread out sample means are around the actual population mean. Imagine you have a lot of small samples, each giving its own mean. When you calculate their spread, you're looking at the standard error. The formula is \( SE = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population standard deviation and \( n \) is the sample size.
If the sample size increases:
  • The denominator \( \sqrt{n} \) becomes larger.
  • This causes the standard error to decrease.
This means sample means are less varied and are closer to the actual population mean. So, increasing the sample size makes your estimate more precise.
Central Limit Theorem
The Central Limit Theorem is a cornerstone of statistics. It tells us how the distribution of sample means behaves. No matter what your original population distribution looks like, if you take large enough samples repeatedly, the distribution of those sample means will look more like a normal distribution. This is really handy because working with a normal distribution is relatively straightforward.
When the sample size increases:
  • The distribution of sample means becomes more normal.
  • The spread or variability decreases, making it narrower.
Even if the population itself is not normally distributed, larger sample sizes help the sample mean distribution to become bell-shaped.
Population Mean Estimation
Estimating the population mean is crucial for data analysis. Using the distribution of sample means, you can make an informed estimate about the population mean. Larger sample sizes significantly impact this estimation.
The benefits of larger sample sizes include:
  • A more accurate and reliable estimation.
  • Reduction in variability as the standard error is smaller.
  • The sampling distribution gets tightly centered around the population mean.
All these factors together ensure that your estimate reflects the actual population mean better, allowing for improved decision-making and analysis based on data.

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Most popular questions from this chapter

At the downtown office of First National Bank there are five tellers. Last week the tellers made the following number of errors each: \(2,3,5,3,\) and \(5 .\) a. How many different samples of 2 tellers are possible? b. List all possible samples of size 2 and compute the mean of each. c. Compute the mean of the sample means and compare it to the population mean.

Suppose we roll a fair die two times. a. How many different samples are there? b. List each of the possible samples and compute the mean. c. On a chart similar to Chart \(8-1\), compare the distribution of sample means with the distribution of the population. d. Compute the mean and the standard deviation of each distribution and compare them.

Suppose your statistics instructor gave six examinations during the semester. You received the following grades (percent correct): 79,64,84,82,92 and 77 . Instead of averaging the six scores, the instructor indicated he would randomly select two grades and compute the final percent correct based on the two percents. a. How many different samples of two test grades are possible? b. List all possible samples of size two and compute the mean of each. c. Compute the mean of the sample means and compare it to the population mean. d. If you were a student, would you like this arrangement? Would the result be different from dropping the lowest score? Write a brief report.

A normal population has a mean of 75 and a standard deviation of \(5 .\) You select a sample of \(40 .\) Compute the probability the sample mean is: a. Less than 74 . b. Between 74 and 76 . c. Between 76 and 77 . d. Greater than 77 .

CRA CDs, Inc., wants the mean lengths of the "cuts" on a CD to be 135 seconds (2 minutes and 15 seconds). This will allow the disk jockeys to have plenty of time for commercials within each 10 -minute segment. Assume the distribution of the length of the cuts follows the normal distribution with a population standard deviation of 8 seconds. Suppose we select a sample of 16 cuts from various CDs sold by CRA CDs, Inc. a. What can we say about the shape of the distribution of the sample mean? b. What is the standard error of the mean? c. What percent of the sample means will be greater than 140 seconds? d. What percent of the sample means will be greater than 128 seconds? e. What percent of the sample means will be greater than 128 but less than 140 seconds?
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