# Sampling distribution exercises and solutions

These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang. Find the mean and standard deviation of the sample mean. Answers S6. Hint: One way to solve the problem is to first find the probability of the complementary event. Many sharks enter a state of tonic immobility when inverted. Borachio eats at the same fast food restaurant every day. If you had this experience, is it particularly strong evidence that the tire is not as good as claimed?

A consumer group buys five such tires and tests them. If the mean is so low, is that particularly strong evidence that the tire is not as good as claimed? Find the indicated probabilities. First verify that the sample is sufficiently large to use the normal distribution. You may assume that the normal distribution applies.

Compute the sample proportion. Suppose this proportion is valid for all homes. Suppose this proportion is valid. Additional Exe rcises Some countries allow individual packages of prepackaged goods to weigh less than what is stated on the package, subject to certain conditions, such as the average of all packages being the stated weight or greater.

An economist wishes to investigate whether people are keeping cars longer now than in the past. Find the sample proportion. Give an interpretation of the result in part b. Is there strong evidence that people are keeping their cars longer than was the case five years ago? A state public health department wishes to investigate the effectiveness of a campaign against smoking.

How strong is the evidence that the campaign to reduce smoking has been effective? How strong is the evidence that the presence of the low-cost clinic has increased the proportion of pet dogs and cats that have been spayed or neutered?

How strong is the evidence that the die is not fair? Rework part a under these circumstances. Give an interpretation of the result in part c. Contributor Anonymous.You may want to use the " r to z' calculator " and the " Calculate Area for a given X " applet for some of these exercises. A population has a mean of 50 and a standard deviation of 6. Given a test that is normally distributed with a mean of and a standard deviation of 12, find:.

What term refers to the standard deviation of the sampling distribution? A questionnaire is developed to assess women's and men's attitudes toward using animals in research. One question asks whether animal research is wrong and is answered on a 7-point scale.

Assume that in the population, the mean for women is 5, the mean for men is 4, and the standard deviation for both groups is 1. Assume the scores are normally distributed. If 12 women and 12 men are selected randomly, what is the probability that the mean of the women will be more than 1.

If the correlation between reading achievement and math achievement in the population of fifth graders were 0.

## 6.E: Sampling Distributions (Exercises)

A normal distribution has a mean of 20 and a standard deviation of Two scores are sampled randomly from the distribution and the second score is subtracted from the first. What august underground snuff edition the probability that the difference score will be greater than 5?

What is the shape of the sampling distribution of r? In what way does the shape depend on the size of the population correlation? If you sample one number from a standard normal distribution, what is the probability it will be 0. A variable is normally distributed with a mean of and a standard deviation of 5. Four scores are randomly sampled. What is the probability that the mean of the four scores is above ? The correlation between self esteem and extraversion is.

A sample of 84 is taken. The standard deviation in both schools is 0. The GPAs of both schools are normally distributed. If 9 students are randomly sampled from each school, what is the probability that:. Suppose 30 people from this city were sampled.

When solving problems where you need the sampling distribution of r, what is the reason for converting from r to z'? In the population, the mean SAT score is Would you be more likely or equally likely to get a sample mean of if you randomly sampled 10 students or if you randomly sampled 30 students? You repeat this process times and plot the distribution of the means. In this case, the sample size is You randomly ask 5 students every day if they watch TV at night. Every day, you would find that 2 of the 5 do watch TV at night.

Suppose in the population, the Anger-Out score for men is two points higher than it is for women. The population variances for men and women are both Assume the Anger-Out scores for both genders are normally distributed.

Given this information about the population parameters:. The following questions use data from the Animal Research AR case study. AR 11 What is the correlation in this sample between the belief that animal research is wrong and belief that animal research is necessary?

Suppose the correlation between the belief that animal research is wrong and the belief that animal research is necessary is .I don't want to reset my password. This textbook survival guide was created for the textbook: Probability and Statistics for Engineers and the Scientists, edition: 9.

This expansive textbook survival guide covers the following chapters and their solutions. Since 76 problems in chapter 8: Fundamental Sampling Distributions and Data Descriptions have been answered, more than students have viewed full step-by-step solutions from this chapter. Chapter 8: Fundamental Sampling Distributions and Data Descriptions includes 76 full step-by-step solutions. A method of variable selection in regression that examines all possible subsets of the candidate regressor variables.

Eficient computer algorithms have been developed for implementing all possible regressions. A study in which a sample from a population is used to make inference to a future population.

Stability needs to be assumed. See Enumerative study. Used to compare hypothesis tests. The ARE of one test relative to another is the limiting ratio of the sample sizes necessary to obtain identical error probabilities for the two procedures. An attribute control chart that plots the total number of defects per unit in a subgroup.

Similar to a defects-per-unit or U chart. The tendency of data to cluster around some value. Central tendency is usually expressed by a measure of location such as the mean, median, or mode. A correction factor used to improve the approximation to binomial probabilities from a normal distribution. In the most general usage, a measure of the interdependence among data. The concept may include more than two variables.

The term is most commonly used in a narrow sense to express the relationship between quantitative variables or ranks. A model to relate a response to one or more regressors or factors that is developed from data obtained from the system. A study in which a sample from a population is used to make inference to the population.

See Analytic study. An analysis of how the variance of the random variable that represents that output of a system depends on the variances of the inputs. A formula exists when the output is a linear function of the inputs and the formula is simpliied if the inputs are assumed to be independent.

About us. Textbook Survival Guides. Elite Notetakers. Referral Program. Campus Marketing Coordinators.I have studied many languages-French, Spanish and a little Italian, but no one told me that Statistics was a foreign language. Sections 4. Kind of like stamp collecting, but with numbers. However, statistics covers much more than that. In fact, descriptive statistics is one of the smallest parts of statistics, and one of the least powerful. The bigger and more useful part of statistics is that it provides tools that let you make inferences about data.

Once you start thinking about statistics in these terms — that statistics is there to help us draw inferences from data — you start seeing examples of it everywhere. A polling company has conducted a survey, usually a pretty big one because they can afford it. For the Federal election, the Australian Electoral Commission reported 4, enrolled voters in New South Whales; so the opinions of the remaining 4, voters about The answer to the question is pretty obvious: if I call people at random, and of them say they intend to vote for the ALP, then it seems very unlikely that these are the only people out of the entire voting public who actually intend to do so.

In other words, we assume that the data collected by the polling company is pretty representative of the population at large. But how representative? At this point everyday intuition starts to break down a bit. We need some more powerful tools than just looking at the numbers and guessing. Inferential statistics provides the tools that we need to answer these sorts of questions, and since these kinds of questions lie at the heart of the scientific enterprise, they take up the lions share of every introductory course on statistics and research methods.

However, our tools for making statistical inferences are 1 built on top of probability theoryand 2 require an understanding of how samples behave when you take them from distributions defined by probability theory…. So, this chapter has two main parts.

A brief introduction to probability theory, and an introduction to sampling from distributions. For example, all of these questions are things you can answer using probability theory:.

Notice that all of these questions have something in common. In the second question, I know that the chance of rolling a 6 on a single die is 1 in 6. In the third question I know that the deck is shuffled properly. And in the fourth question, I know that the lottery follows specific rules. You get the idea. The critical point is that probabilistic questions start with a known model of the world, and we use that model to do some calculations.

The underlying model can be quite simple. What about statistics? Statistical questions work the other way around. In statistics, we know the truth about the world. All we have is the data, and it is from walkatha pdf data that we want to learn the truth about the world. Statistical questions tend to look more like these:.

If five cards off the top of the deck are all hearts, how likely is it that the deck was shuffled? This time around, the only thing we have are data. What I know is that I saw my friend flip the coin 10 times and it came up heads every time.

And what I want to infer is whether or not I should conclude that what I just saw was actually a fair coin being flipped 10 times in a row, or whether I should suspect that my friend is playing a trick on me. The data I have look like this:. If the coin is fair, then the model I should adopt is one that says that the probability of heads is 0.Central limit theorem wikipedialookup.

Sign in Sign up. Thank you for your participation! Document related concepts. A bottling company uses a filling machine to fill plastic bottles with a popular cola. The bottles are supposed to contain milliliters. What is the probability that an individual bottle contains less than ml? What is the probability that the mean contents of the bottles in a six-pack is less than ml?

The level of nitrogen oxide NOX in the exhaust of a particular car model varies with mean 1. A company has cars of this model in its fleet. The number of flaws per square yard in a type of carpet material varies, with mean 1.

The distribution is not normal--in fact, it is discrete.

## Chapter 8: Fundamental Sampling Distributions and Data Descriptions:

An inspector studies random square yards of the material, records the number of flaws found in each square yard, and calculates the mean number of flaws per square yard inspected.

Use the central limit theorem to find the approximate probability that the mean number of flaws exceeds 2 per square yard. The number of accidents per week at a hazardous intersection varies with mean 2. This distribution is discrete and so is certainly not normal. What is the approximate probability that there are fewer than accidents at the intersection in a year?

Solutions 1a. So, 1. Related documents.The sampling distribution can be described by calculating its mean and standard error. The central limit theorem states that if the sample is large enough, its distribution will approximate that of the population you took the sample from. This means that if the population had a normal distribution, so will the sample. If you do not know the population distribution, it is generally assumed to be normal.

You will need to know the standard deviation of the population in order to calculate the sampling distribution. Add all of the observations together and then divide by the total number of observations in the sample.

For example, a sample of heights of everyone in a town might have observations of 60 inches, 64 inches, 62 inches, 70 inches and 68 inches and the town is known to have a normal height distribution and standard deviation of 4 inches in its heights. If the population size is very large, all the people in a city for example, you need only divide 1 by the sample size. Take the square root of the result from Step 2 and then multiply it by the standard deviation of the population.

For the example, the square root of 0. Then, 0. The sample's standard error is 1. Together, the mean, The sample has a normal distribution because the town does. Kaylee Finn began writing professionally for various websites inprimarily contributing articles covering topics in business personal finance. She brings expertise in the areas of taxes, student loans and debt management to her writing.

How to Calculate X-bar. How to Calculate Skew. How do I Calculate Cumulative Percentages? How to Compute a Population Mean.

## Sampling distribution of a sample mean example

How to Calculate the Distribution of the Mean. How to Calculate a T-Statistic. How to Estimate the True Proportion. How to Calculate the Coefficient of Variation. How to Calculate Sample Mean. How to Calculate Relative Standard Error. How to Calculate Mean Deviation.

Slovin's Formula Sampling Techniques.A statistic, such as the sample mean or the sample standard deviation, is a number computed from a sample. Since a sample is random, every statistic is a random variable: it varies from sample to sample in a way that cannot be predicted with certainty. As a random variable it has a mean, a standard deviation, and a probability distribution.

The probability distribution of a statistic is called its sampling distribution The probability distribution of a sample statistic when the statistic is viewed as a random variable. Typically sample statistics are not ends in themselves, but are computed in order to estimate the corresponding population parameters, as illustrated in the grand picture of statistics presented in Figure 1. This chapter introduces the concepts of the mean, the standard deviation, and the sampling distribution of a sample statistic, with an emphasis on the sample mean x.

In actual practice we would typically take just one indus pharma lahore location. Imagine however that we take sample after sample, all of the same size nand compute the sample mean x - of each one. We will likely get a different value of x - each time. The sample mean x - is a random variable: it varies from sample to sample in a way that cannot be predicted with certainty.

We will write X - when the sample mean is thought of as a random variable, and write x balterio oak laminate flooring for the values that it takes.

The random variable X - has a mean The number about which means computed from samples of the same size center. Here is an example with such a small population and small sample size that we can actually write down every single sample. A rowing team consists of four rowers who weigh,and pounds. Find all possible random samples with replacement of size two and compute the sample mean for each one. Use them to find the probability distribution, the mean, and the standard deviation of the sample mean X.

The following table shows all possible samples with replacement of size two, along with the mean of each:. The table shows that there are seven possible values of the sample mean X. Since the 16 samples are equally likely, we obtain the probability distribution of the sample mean just by counting:.

Now we apply the formulas from Section 4. The mean of the sample mean X - that we have just computed is exactly the mean of the population. These relationships are not coincidences, but are illustrations of the following formulas. The second formula says that averages computed from samples vary less than individual measurements on the population do, and quantifies the relationship.

Suppose random samples of size are drawn from the population of vehicles. Random samples of size are drawn from a population with mean and standard deviation Find the mean and standard deviation of the sample mean.

Random samples of size 64 are drawn from a population with mean 32 and standard deviation 5. A population has mean 75 and standard deviation A population has mean 5. In Note 6. The probability distribution is:.

Figure 6. Whereas the distribution of the population is uniform, the sampling distribution of the mean has a shape approaching the shape of the familiar bell curve.

This phenomenon of the sampling distribution of the mean taking on a bell shape even though the population distribution is not bell-shaped happens in general. Here is a somewhat more realistic example. Suppose we take samples of size 1, 5, 10, or 20 from a population that consists entirely of the numbers 0 and 1, half the population 0, half 1, so that the population mean is 0.

The sampling distributions are:. These are homework exercises to accompany the Textmap created for "Introductory The Sampling Distribution of the Sample Mean. Use them to find the probability distribution, the mean, and the standard deviation of the sample mean ˉX. Solution. The following table shows all possible. No, the previous answer only depended on the standard deviation of the sampling distribution of the sample mean, not the mean itself.

a. From Exercise 2) Explain the sampling distribution of the sample Example. Do problems 10 and 12 Solution. The mean commuting time of this population is.

Elementary Statistics—Sampling-Distribution Exercises but it's important that you sketch your solutions (to refer back to when studying). All material presented in the Sampling Distributions chapter (a) What are the mean and standard deviation of the sampling distribution of the mean for N. Solutions 3/5 [email protected] Exercise 1.

For the population of individuals who own an sampling distribution of the sample proportion. Test your understanding with practice problems and step-by-step solutions. Browse through all study tools. Question & Answers. Sampling distribution is for all possible samples of size n. element population of Xi's, in problems where there are many.

Using the data from Exercisefor a sample of 25 female students, calcu-late the standard error of the mean, draw the sampling distribution about μ. Here's the type of problem you might see on the AP Statistics exam where you have to use the sampling distribution of a sample mean.

Which of the following sample sizes would have the lowest variability in the sampling distribution? a. 10 b. c. ANSWER: c (). #4.