/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Consider an \(x\) distribution w... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 8

Consider an \(x\) distribution with standard deviation \(\sigma=12\). (a) If specifications for a research project require the standard error of the corresponding \(\bar{x}\) distribution to be 2, how large does the sample size need to be? (b) If specifications for a research project require the standard error of the corresponding \(\bar{x}\) distribution to be 1, how large does the sample size need to be?

Short Answer

Expert verified
(a) 36; (b) 144

Step by step solution

01

Understand the Relationship

The standard error (SE) of the mean is given by the formula: \( SE = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population standard deviation and \( n \) is the sample size. We need to solve for \( n \).
02

Rearrange the Formula

To find the sample size \( n \), rearrange the formula: \( n = \left( \frac{\sigma}{SE} \right)^2 \). This formula shows that the sample size is the square of the ratio of the population standard deviation to the standard error.
03

Calculate for Part (a)

For part (a), where \( SE = 2 \), substitute the values into the formula: \( n = \left( \frac{12}{2} \right)^2 = 6^2 = 36 \). The sample size required for a standard error of 2 is 36.
04

Calculate for Part (b)

For part (b), where \( SE = 1 \), substitute the values into the formula: \( n = \left( \frac{12}{1} \right)^2 = 12^2 = 144 \). The sample size required for a standard error of 1 is 144.

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Key Concepts

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

Sample Size Calculation
Determining the correct sample size is crucial in statistical studies. It directly affects the reliability and validity of the research outcomes.To calculate the sample size required for a study, you can use the formula for the standard error (SE) of the mean: \[ SE = \frac{\sigma}{\sqrt{n}} \] By rearranging this equation, we find the formula to determine sample size: \[ n = \left( \frac{\sigma}{SE} \right)^2 \] This shows that sample size is the square of the ratio of the population standard deviation to the desired standard error.
  • A smaller standard error requires a larger sample size.
  • The formula indicates a direct relationship between the desired precision (as reflected in a smaller SE) and the number of observations needed (sample size).
Reaching the correct sample size ensures that the study鈥檚 results are statistically significant and that they truly represent the population.
Population Standard Deviation
The population standard deviation, represented as \( \sigma \), measures the spread of data points around the mean in a population. It is a critical element in various statistical calculations.
  • This statistic is crucial when determining how data points deviate from the mean.
  • In many real-world scenarios, it isn鈥檛 feasible to calculate a true standard deviation across a whole population, so a representative sample is often used.
In sample size calculations, the population standard deviation helps in determining how large a sample should be to achieve a certain level of accuracy. Particularly, when given the standard deviation, one can calculate how much the sample mean will likely fluctuate from that of the population mean.
Sampling Distribution
The concept of sampling distribution is central to understanding statistics. It refers to the probability distribution of a statistic obtained from a larger number of samples drawn from a specific population. There are some key ideas related to sampling distribution:
  • The mean of the sampling distribution of the sample mean is equal to the population mean.
  • The spread of the sampling distribution is determined by the standard error, which is affected by the sample size.
  • As the sample size increases, the sampling distribution becomes more narrow, providing a more precise estimate of the population parameter.
This concept underscores the importance of sample size, as it directly impacts the variability of sampling distributions and the reliability of conclusions drawn from statistical data.
Statistics Education
Statistics education is pivotal in helping individuals interpret data and make informed decisions. It goes beyond just number-crunching and delves into the understanding of context and variability. Key aspects to focus on include:
  • Grasping foundational concepts like mean, median, mode, variance, and standard deviation.
  • Understanding the role of randomness and sample variation in statistics.
  • Learning how to apply statistical techniques correctly and understanding the impact these techniques have on data interpretation.
Solid statistics education empowers students to apply analytical thinking to real-life issues, fostering a deeper understanding of how conclusions are drawn from data and the potential consequences of these conclusions.

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Most popular questions from this chapter

What is the formula for the standard error of the normal approximation to the \(\hat{p}\) distribution? What is the mean of the \(\hat{p}\) distribution?

What is the standard deviation of a sampling distribution called?

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.

Expand Your Knowledge: Totals Instead of Averages Let \(x\) be a random variable that represents checkout time (time spent in the actual checkout process) in minutes in the express lane of a large grocery. Based on a consumer survey, the mean of the \(x\) distribution is about \(\mu=2.7\) minutes, with standard deviation \(\sigma\) \(=0.6\) minute. Assume that the express lane always has customers waiting to be checked out and that the distribution of \(x\) values is more or less symmetrical and mound-shaped. What is the probability that the total checkout time for the next 30 customers is less than 90 minutes? Let us solve this problem in steps. (a) Let \(x_{i}\) (for \(\left.i=1,2,3, \ldots, 30\right)\) represent the checkout time for each customer. For example, \(x_{1}\) is the checkout time for the first customer, \(x_{2}\) is the checkout time for the second customer, and so forth. Each \(x_{i}\) has mean \(\mu=2.7\) minutes and standard deviation \(\sigma=0.6\) minute. Let \(w=x_{1}+x_{2}+\cdots+x_{30 .}\) Explain why the problem is asking us to compute the probability that \(w\) is less than 90 . (b) Use a little algebra and explain why \(w<90\) is mathematically equivalent to \(w / 30<3 .\) Since \(w\) is the total of the \(30 x\) values, then \(w / 30=\bar{x}\). Therefore, the statement \(\bar{x}<3\) is equivalent to the statement \(w<90\). From this we conclude that the probabilities \(P(\bar{x}<3)\) and \(P(w<90)\) are equal. (c) What does the central limit theorem say about the probability distribution of \(\bar{x}\) ? Is it approximately normal? What are the mean and standard deviation of the \(\bar{x}\) distribution? (d) Use the result of part \((c)\) to compute \(P(\bar{x}<3)\). What does this result tell you about \(P(w<90)\) ?

What is a random sample from a population? (Hint: See Section 1.2.)
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