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Chapter 7: Problem 30

A population has a normal distribution. \(\Lambda\) sample of size \(n\) is selected from this population. Describe the shape of the sampling distribution of the sample mean for each of the following cases. a. \(n=94\) b. \(n=11\)

Short Answer

Expert verified
For the case a. with n=94, the shape of the sampling distribution of the sample mean will be approximately normal because n>30. In the case b. with n=11, without additional information about the population, it can't be definitively stated.

Step by step solution

01

Identify & Analyze

Identify that the task is about the shape of the sampling distribution of the sample mean, which depends on the size of the sample (n). The key is realizing that, according to the Central Limit Theorem (CLT), any sample with a size above 30 will have a sampling distribution of the mean that is approximately normal.
02

Case a. n=94

Given that n=94 which is greater than 30, so according to the Central Limit theorem, the shape of the sampling distribution of the sample mean will be approximately normal.
03

Case b. n=11

Given that n=11 which is less than 30, since we are not provided with any additional information about the population (e.g., is it symmetrical, skewed, etc.?), we cannot make any definitive statement about the shape of the sampling distribution of the mean. However, it's important to remember in practice, many distributions are close enough to the normal distribution when the sample size is reasonably large, even if the population distribution is not perfectly normal.

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Key Concepts

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

Sampling Distribution
When working with statistics, the concept of sampling distribution is crucial for understanding how sample means behave across different samples. If you take multiple samples from a population and calculate their means, these means form what we call a sampling distribution.
This distribution of sample means allows us to make inferences about the population mean.
  • The sampling distribution of the sample mean becomes approximately normal if the sample size is large enough, thanks to the Central Limit Theorem (CLT).
  • This is beneficial when analyzing data, as it enables us to use properties of the normal distribution to make probability-based inferences.
This foundational concept enables deeper analysis and helps statisticians predict behaviors even if they cannot test the entire population directly.
Sample Size
Sample size, often denoted as "n," is a key factor in determining the behavior of the sampling distribution. A larger sample size gives more accurate estimates of the population parameters.
  • According to the Central Limit Theorem, a sample size of over 30 usually means the sampling distribution of the sample mean will be approximately normal.
  • Smaller sample sizes, like those less than 30, may not form a sampling distribution that is normally distributed unless the population from which they were drawn is normal.
Thus, the choice of sample size can greatly impact the reliability and accuracy of statistical conclusions, and larger samples often lead to more reliable outcomes.
Normal Distribution
The normal distribution is a fundamental concept in statistics, characterized by its bell-shaped curve. It is symmetric around the mean, which implies that data near the mean are more frequent in occurrence than data far from the mean.
  • The normal distribution has important properties: 68% of the data lies within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations of the mean.
  • In the context of CLT, when sample sizes are large, the distribution of sample means will approximate this shape.
This property makes it easier to conduct hypothesis tests and calculate probabilities, providing a solid foundation for various statistical techniques.
Population Distribution
Population distribution refers to the distribution of all the values of a characteristic in an entire population. It shows how values are spread out over the entire population.
  • The shape of the population distribution can provide insights when analyzing sample data. In cases where the population distribution is known to be normal, understanding the sample means becomes much simpler.
  • Even if the population distribution is not normal, the CLT suggests that the sampling distribution of the sample mean will still be approximately normal for large sample sizes.
Knowing the population distribution can greatly aid in predicting behaviors and outcomes, thereby enabling better statistical analysis and decisions.

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Most popular questions from this chapter

A television reporter is covering the election for mayor of a large city and will conduct an exit poll (interviews with voters immediately after they vote) to make an early prediction of the outcome. Assume that the eventual winner of the election will get \(60 \%\) of the votes. a. What is the probability that a prediction based on an exit poll of a random sample of 25 voters will be correct? In other words, what is the probability that 13 or more of the 25 voters in the sample will have voted for the eventual winner? b. How large a sample would the reporter have to take so that the probability of correctly predicting the outcome would be 95 or higher?

quarter), Britons spend an av鈥� # According to a 2004 survey by the telecommunications division of British Gas (Source: http://www.literacytrust.org.uk/Database/texting html#quarter), Britons spend an average of 225 minutes per day communicating electronically (on a landline phone, on a mobile phone, by emailing, or by texting). Assume that currently such communication times for all Britons are normally distributed with a mean of 225 minutes per day and a standard deviation of 62 minutes per day. Let \(\bar{x}\) be the average time spent per day communicating electronically by 20 randomly selected Britons. Find the mean and the standard deviation of the sampling distribution of \(\bar{x}\). What is the shape of the sampling distribution of \(\bar{x}\) ?

In an article by Laroche et al. (The Journal of the American Board of Family Medicine \(2007 ; 20: 9-15\) ), the average daily fat intake of U.S. adults with children in the household is \(91.4\) grams, with a standard deviation of \(93.25\) grams. These results are based on a sample of 3714 adults. Suppose that these results hold true for the current population distribution of daily fat intake of such adults, and that this distribution is strongly skewed to the right. Let \(\bar{x}\) be the average daily fat intake of 20 randomly selected U.S. adults with children in the household. Find the mean and the standard deviation of the sampling distribution of \(\bar{x}\). Do the same for a random sample of size \(75 .\) How do the shapes of the sampling distributions differ for the two sample sizes?

Mong Corporation makes auto batteries. The company claims that \(80 \%\) of its LL 70 batteries are good for 70 months or longer. Assume that this claim is true. Iet \(\hat{p}\) be the proportion in a sample of 100 such batteries that are good for 70 months or longer. a. What is the probability that this sample proportion is within \(.05\) of the population proportion? b. What is the probability that this sample proportion is less than the population proportion by .06 or more? c. What is the probability that this sample proportion is greater than the population proportion by \(.07\) or more?

If all possible samples of the same (large) size are selected from a population, what percentage of all the sample means will be within \(1.5\) standard deviations of the population mean?
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