/*! 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 20 If a population has a standard d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 20

If a population has a standard deviation \(\sigma\) of 25 units, what is the standard error of the mean if samples of size 16 are selected? Samples of size \(36 ?\) Samples of size \(100 ?\)

Short Answer

Expert verified
The standard error of the mean for sample sizes 16, 36 and 100 are approximately 6.25, 4.16 and 2.5 units respectively.

Step by step solution

01

Calculating the standard error for sample size 16

Given \(\sigma = 25\) units and \(n = 16\), substitute these values into the formula \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\). The standard error for sample size 16 will be \(\frac{25}{\sqrt{16}}\).
02

Calculating the standard error for sample size 36

For a sample size of \(n = 36\), substitute this into the formula to calculate the standard error. The standard error for sample size 36 will be \(\frac{25}{\sqrt{36}}\).
03

Calculating the standard error for sample size 100

Finally, for a sample size of \(n = 100\), substituting into the formula gives the standard error for this sample size as \(\frac{25}{\sqrt{100}}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Sample Size
The sample size, often denoted by \( n \), is a critical component in determining the properties of a sample. In statistics, the sample size refers to the number of observations or data points collected from a larger population to represent that population. The choice of sample size affects the precision of estimates. More observations generally lead to more reliable statistical analyses, as a greater sample size tends to reduce the uncertainty.
  • A larger sample size results in a smaller standard error, indicating more precise sample mean estimates.
  • A smaller sample, while easier to manage, may lead to less accurate representations of the population.
  • The formula for calculating standard error \( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \) shows a clear dependency on the sample size \( n \).
Therefore, when analyzing data, it鈥檚 vital to carefully consider the sample size to balance practical constraints with the statistical accuracy needed for the study.
Standard Deviation
Standard deviation, represented by the symbol \( \sigma \), is a key statistics concept that measures the amount of variation or dispersion in a set of values. It provides insight into how much the individual data points differ from the mean, or average, of the dataset.
The standard deviation is significant because it gives context to how spread out the values are.
  • A small standard deviation indicates that the data points are close to the mean.
  • A large standard deviation shows that the data points are more spread out over a wide range of values.
  • This statistic helps to understand how much uncertainty or variability exists within a population.
Understanding the standard deviation is essential for calculating the standard error of the mean, which shows how much the sample mean is expected to fluctuate from the true population mean.
Population Statistics
Population statistics involve studying the characteristics of a large group, or population, to make inferences about the group's properties. These statistics are used to summarize and analyze the collective data to understand patterns or trends within it.
In this context:
  • Population statistics can include data like the mean (average), variance, and standard deviation.
  • The population's standard deviation is used to calculate the standard error, providing insight into how sample means might vary if different samples are taken from the same population.
  • By knowing these characteristics, researchers can make informed inferences and decisions based on sample data.
Using population statistics, especially standard deviation, helps in determining how representative a sample's statistics are of the entire population, thereby guiding correct interpretations and conclusions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A researcher wants to take a simple random sample of about \(5 \%\) of the student body at each of two schools. The university has approximately 20,000 students, and the college has about 5000 students. Identify each of the following as true or false and justify your answer. a. The sampling variability is the same for both schools. b. The sampling variability for the university is higher than that for the college. c. The sampling variability for the university is lower than that for the college. d. No conclusion about the sampling variability can be stated without knowing the results of the study.

a. Use a computer to randomly select 100 samples of size 6 from a normal population with mean \(\mu=20\) and standard deviation \(\sigma=4.5\). b. Find mean \(\bar{x}\) for each of the 100 samples. c. Using the 100 sample means, construct a histogram, find mean \(\bar{x},\) and find the standard deviation \(s_{\bar{x}}\). MINITAB a. Use the Normal RANDOM DATA commands on page 91 , replacing generate with 100 , store in with \(\mathrm{Cl}-\mathrm{C} 6\), mean with \(20,\) and standard deviation with \(4.5 .\) b. Use the ROW STATISTICS commands on page 318 , replacing input variables with \(\mathrm{C} 1-\mathrm{C} 6\) and store result in with C7. c. Use the HISTOGRAM commands on page 53 for the data in C7. To adjust the histogram, select Binning with midpoint and midpoint positions \(12.8: 27.2 / 1.8 .\) Use the MEAN and STANDARD DEVIATION commands on pages 65 and 79 for the data in C7. Excel a. Use the Normal RANDOM NUMBER GENERATION commands on page \(91,\) replacing number of variables with 6 number of random numbers with \(100,\) mean with \(20,\) and standard deviation with 4.5 b. Activate cell G1. Choose: \(\quad\) Insert function, \(f_{x}>\) Statistical \(>\) AVERAGE \(>\mathrm{OK}\) Enter: \(\quad\) Number1: (A1:F1 or select cells) Drag: Bottom right corner of average value box down to give other averages c. Use the RANDOM NUMBER GENERATION Patterned Distribution commands in Exercise 6.71 (a) on page 291 replacing the first value with 12.8 , the last value with \(27.2,\) the steps with \(1.8,\) and the output range with H1. Use the HISTOGRAM commands on pages \(53-54\) with column \(G\) as the input range and column \(H\) as the bin range. Use the MEAN and STANDARD DEVIATION commands on pages 65 and 79 for the data in column G. TI-83/84 Plus a. Use the Normal RANDOM DATA and STO commands on page \(91,\) replacing Enter with 20,4.5,100 ). Repeat the preceding commands five more fimes, storing data in \(L 2\) \(13,14,15,\) and \(L 6,\) respectively. b. Enter: \(\quad(\mathrm{L} 1+\mathrm{L} 2+\mathrm{L} 3+\mathrm{L} 4+\mathrm{L} 5+\mathrm{L} 6) / 6\) Choose: \(\quad \mathrm{STO} \rightarrow \mathrm{L} 7\) (use ALPHA key for the "L" or use "MEAN"? c. Choose: \(\quad 2 \mathrm{nd}>\) STAT \(\mathrm{PLOT}>1:\) Plot 1 Choose: Window Enter: \(\quad 12.8,27.2,1.8,0,40,5,1\) Choose: \(\quad\) Trace \(>>>\) Choose: \(\quad\) STAT \(>\) CALC \(>\) 1:1-VAR STATS \(>2 \mathrm{nd}>\) LIST Select: L7. d. Compare the results of part c with the three statements made in the SDSM.

Consider the set of odd single-digit integers $$\\{1,3,5,7,9\\}$$. a. Make a list of all samples of size 2 that can be drawn from this set of integers. (Sample with replacement; that is, the first number is drawn, observed, and then replaced [returned to the sample set] before the next drawing.) b. Construct the sampling distribution of sample means for samples of size 2 selected from this set. c. Construct the sampling distributions of sample ranges for samples of size 2.

TIMSS 2007 (Trends in International Mathematics and Science Study) focused on the mathematics and science achievement of eighth-grade students throughout the world. A total of 8 countries (including the United States) participated in the study. The mean math exam score for U.S. students was 509 with a standard deviation of \(88 .\) Assuming the scores are normally distributed, find the following for a sample of 150 students. a. Find the probability that the mean TIMSS score for a randomly selected group of eighth-grade students would be between 495 and 515. b. Find the probability that the mean TIMSS score for a randomly selected group of eighth-grade students would be less than 520. c. Do you think the assumption of normality is reasonable? Explain.

Using a set of five dice, roll the dice and determine the mean number of dots showing on the five dice. Repeat the experiment until you have 25 sample means. a. Draw a dotplot showing the distribution of the 25 sample means. (See Example \(7.2,\) p. \(315 .\) ) b. \(\quad\) Describe the distribution of \(\bar{x}\) 's in part a. c. \(\quad\) Repeat the experiment to obtain 25 more sample means and add these \(25 . \bar{x}\) 's to your dotplot. Describe the distribution of 50 means.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.