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Chapter 6: Problem 9

Consider two \(\bar{x}\) distributions corresponding to the same \(x\) distribution. The first \(\bar{x}\) distribution is based on samples of size \(n=100\) and the second is based on samples of size \(n=225 .\) Which \(\bar{x}\) distribution has the smaller standard error? Explain.

Short Answer

Expert verified
The second distribution with \( n = 225 \) has the smaller standard error.

Step by step solution

01

Understanding the Problem

We need to determine which of the two sample means, based on different sample sizes, has the smaller standard error of the mean. The standard error of the mean measures the dispersion of sample mean estimates from the true population mean.
02

Formula for Standard Error of the Mean

The formula for the standard error of the mean (SEM) is given by \( SEM = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the standard deviation of the population and \( n \) is the sample size.
03

Calculate First Distribution's Standard Error

For the first distribution, where \( n = 100 \), the standard error is \( \frac{\sigma}{\sqrt{100}} = \frac{\sigma}{10} \).
04

Calculate Second Distribution's Standard Error

For the second distribution, where \( n = 225 \), the standard error is \( \frac{\sigma}{\sqrt{225}} = \frac{\sigma}{15} \).
05

Comparison of the Standard Errors

The standard error for the distribution with \( n = 225 \) is \( \frac{\sigma}{15} \), which is smaller than the standard error for \( n = 100 \), \( \frac{\sigma}{10} \), because \( \frac{1}{15} < \frac{1}{10} \). Therefore, the larger sample size results in a smaller standard error.

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Key Concepts

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

Sample Size
Sample size refers to the number of observations or data points included in a sample drawn from a larger population. It is an essential aspect of statistical analysis because it can significantly impact the reliability and precision of your estimates.
The larger the sample size ( ), the more information you gather, which usually leads to more accurate estimates of the population parameters.
  • Less Error: Larger sample sizes typically result in smaller standard errors. This means the sample mean will be closer to the population mean.
  • Representation: Larger samples tend to provide a more accurate representation of the population, thereby reducing the effects of random error.
  • Stability: Larger samples reduce the effect of outliers, leading to more stable results.
When conducting any analysis, choosing an appropriate sample size is crucial to ensure your findings are both valid and reliable.
Sampling Distribution
A sampling distribution is the probability distribution of a given statistic based on a random sample. It's a critical concept in inferential statistics, allowing us to make decisions about the population based on sample data.
Here's what you need to know about sampling distributions:
  • Predictable Behavior: As the sample size increases, the sampling distribution of the sample mean becomes approximately normally distributed due to the Central Limit Theorem.
  • Mean and Variance: The mean of a sampling distribution of the mean is equal to the population mean () and the variance is given by /n.
  • Standard Error: The standard deviation of the sampling distribution is known as the standard error of the mean, providing a measure of the typical distance between the sample mean and the population mean.
Understanding sampling distributions is vital as they form the basis for hypothesis testing and estimation techniques.
Standard Deviation
Standard deviation is a measure that indicates the amount of variation or dispersion present in a set of values. In the context of this problem, it represents the variation within the population from which samples are drawn. A few core points about standard deviation include:
  • Pivotal Role: It is central to determining the standard error of the mean, as seen in the formula:
    \[ SEM = \frac{\sigma}{\sqrt{n}} \]
    where \( \sigma \) is the standard deviation of the population.
  • Interpretation: Larger standard deviations suggest more spread in the data, whereas smaller standard deviations indicate that data points are more clustered around the mean.
  • Relation to Data: It's a critical statistic because it provides insight into the variability and helps quantify risk or uncertainty in statistical findings.
The standard deviation is essential when you aim to understand the spread of data in your sample or the larger population.
Population Mean
The population mean, denoted as \( \mu \), is the average of all data points in a population. It's a fundamental measure in statistics, indicating the central tendency of the population data.
Here's why it's important:
  • Benchmark: The population mean serves as a benchmark for assessing sample means, allowing us to determine how close these means are to the actual population parameter.
  • Comparison: Knowing the population mean helps in comparing different samples or populations. It gives a point of reference in hypothesis testing and confidence interval estimation.
  • True Measure: While often unknown and estimated via sample data, the population mean is the true measure we aim to approximate through statistical analysis.
In statistical inference, the ultimate goal is often to make conclusions about the population mean using sample data, making it a key component of many analyses.

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Most popular questions from this chapter

Sketch the areas under the standard normal curve over the indicated intervals and find the specified areas. To the right of \(z=-1.22\)

Find the \(z\) value described and sketch the area described.Find \(z\) such that \(82 \%\) of the standard normal curve lies to the right of \(z\).

Templeton World is a mutual fund that invests in both U.S. and foreign markets. Let \(x\) be a random variable that represents the monthly percentage return for the Templeton World fund. Based on information from the Morningstar Guide to Mutual Funds (available in most libraries), \(x\) has mean \(\mu=1.6 \%\) and standard deviation \(\sigma=0.9 \%\) (a) Templeton World fund has over 250 stocks that combine together to give the overall monthly percentage return \(x .\) We can consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all world stocks. Then we see that the overall monthly return \(x\) for Templeton World fund is itself an average return computed using all 250 stocks in the fund. Why would this indicate that \(x\) has an approximately normal distribution? Explain. Hint: See the discussion after Theorem 6.2. (b) After 6 months, what is the probability that the average monthly percentage return \(\bar{x}\) will be between \(1 \%\) and \(2 \% ?\) Hint: See Theorem \(6.1,\) and assume that \(x\) has a normal distribution as based on part (a). (c) After 2 years, what is the probability that \(\bar{x}\) will be between \(1 \%\) and \(2 \% ?\) (d) Compare your answers to parts (b) and (c). Did the probability increase as \(n\) (number of months) increased? Why would this happen? (e) Interpretation: If after 2 years the average monthly percentage return \(\bar{x}\) was less than \(1 \%,\) would that tend to shake your confidence in the statement that \(\mu=1.6 \% ?\) Might you suspect that \(\mu\) has slipped below \(1.6 \% ?\) Explain.

Sketch the areas under the standard normal curve over the indicated intervals and find the specified areas. Between \(z=0\) and \(z=-1.93\)

A person's blood glucose level and diabetes are closely related. Let \(x\) be a random variable measured in milligrams of glucose per deciliter \((1 / 10 \text { of a liter })\) of blood. After a 12 -hour fast, the random variable \(x\) will have a distribution that is approximately normal with mean \(\mu=85\) and standard deviation \(\sigma=25\) (Source: Diagnostic Tests with Nursing Implications, edited by S. Loeb, Springhouse Press). Note: After 50 years of age, both the mean and standard deviation tend to increase. What is the probability that, for an adult (under 50 years old) after a 12 -hour fast, (a) \(x\) is more than \(60 ?\) (b) \(x\) is less than \(110 ?\) (c) \(x\) is between 60 and \(110 ?\) (d) \(x\) is greater than 125 (borderline diabetes starts at 125 )?
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