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Chapter 9: Problem 23

Which distribution do you use when the standard deviation is not known and you are testing one population mean? Assume sample size is large.

Short Answer

Expert verified
Use the normal distribution (z-distribution) for large samples when standard deviation is unknown.

Step by step solution

01

Identify the Known and Unknown Elements

We know the population mean is being tested and the standard deviation is not provided. Moreover, the sample size is indicated as large, which typically means more than 30.
02

Determine the Appropriate Distribution

When the sample size is large and the population standard deviation is unknown, the sample size itself justifies using the normal distribution according to Central Limit Theorem properties.
03

Use of the t-distribution in Context

However, traditionally, if the standard deviation is unknown, one might consider using the t-distribution. In large samples, the t-distribution approaches the normal distribution, so this is typically why the z-distribution is commonly used when the sample size exceeds 30.
04

Final Decision Based on Conditions

Given the information, because we have a large sample size, the standard practice would recommend using the z-distribution for hypothesis testing, despite the unknown standard deviation, due to the sample size's mitigating effect.

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Key Concepts

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

Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental principle in statistics that helps us understand the behavior of sample means. It states that, no matter the distribution of the population, the distribution of the sample means will approximate a normal distribution as the sample size becomes larger.
This is especially true when the sample size is greater than 30. The CLT enables statisticians to make inferences about population means and test hypotheses, even if they don't know the population's original distribution. This theorem is crucial in deciding which statistical methods and distributions to apply, such as using the normal distribution for large samples.
Population Mean
The population mean is the average of all the values in a population. It's a measure of the central tendency, indicating where the middle of the data set lies. When conducting statistical tests, especially when the goal is to understand a parameter like the population mean, it becomes crucial.
Often, the true population mean is unknown, so statisticians use the sample mean, which is calculated from collected samples, as an estimate. Understanding how close the sample mean is to the population mean is aided by concepts like the Central Limit Theorem and standard error.
Sample Size
Sample size is the number of observations in a sample and plays a critical role in statistics. A larger sample size generally yields more reliable estimates of the population parameters, such as the mean.
Large sample sizes (typically those over 30) allow us to utilize the Central Limit Theorem, which supports the use of the normal distribution when testing hypotheses, even when the population's standard deviation is unknown.
  • Smaller samples may not accurately reflect the population and might require the use of other distributions, such as the t-distribution.
  • Larger samples help reduce the margin of error and increase the power of a test.
Normal Distribution
Normal distribution is known for its bell-shaped curve and is essential in statistics due to its properties. When a variable is normally distributed, it means that most observations cluster around the mean and the probabilities for values are symmetrically distributed on either side.
The Central Limit Theorem allows us to use normal distribution for inference about sample means, assuming a large enough sample size.
  • In practice, when the sample size is large, the normal distribution simplifies the calculations necessary for hypothesis testing.
  • Even if the population's distribution is unknown or not normal at the outset, the sample means can approximate a normal distribution under large sample conditions.
This makes it a cornerstone in statistical reasoning and a reliable tool for hypothesis testing.

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Most popular questions from this chapter

Instructions: For the following ten exercises, Hypothesis testing: For the following ten exercises, answer each question. a. State the null and alternate hypothesis. b. State the p-value. c. State alpha. d. What is your decision? e. Write a conclusion. f. Answer any other questions asked in the problem. According to the Center for Disease Control website, in 2011 at least 18% of high school students have smoked a cigarette. An Introduction to Statistics class in Davies County, KY conducted a hypothesis test at the local high school (a medium sized鈥揳pproximately 1,200 students鈥搒mall city demographic) to determine if the local high school鈥檚 percentage was lower. One hundred fifty students were chosen at random and surveyed. Of the 150 students surveyed, 82 have smoked. Use a significance level of 0.05 and using appropriate statistical evidence, conduct a hypothesis test and state the conclusions.

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. State the null and alternative hypotheses.

The mean work week for engineers in a start-up company is believed to be about 60 hours. A newly hired engineer hopes that it's shorter. She asks ten engineering friends in start-ups for the lengths of their mean work weeks. Based on the results that follow, should she count on the mean work week to be shorter than 60 hours? Data (length of mean work week): 70; 45; 55; 60; 65; 55; 55; 60; 50; 55.

You are performing a hypothesis testof a single population proportion. You find out that \(n p\) is less than five. What must you do to be able to perform a valid hypothesis test?

"Dalmatian Darnation," by Kathy Sparling A greedy dog breeder named Spreckles Bred puppies with numerous freckles The Dalmatians he sought Possessed spot upon spot The more spots, he thought, the more shekels. His competitors did not agree That freckles would increase the fee. They said, 鈥淪pots are quite nice But they don't affect price; One should breed for improved pedigree.鈥� The breeders decided to prove This strategy was a wrong move. Breeding only for spots Would wreak havoc, they thought. His theory they want to disprove. They proposed a contest to Spreckles Comparing dog prices to freckles. In records they looked up One hundred one pups: Dalmatians that fetched the most shekels. They asked Mr. Spreckles to name An average spot count he'd claim To bring in big bucks. Said Spreckles, 鈥淲ell, shucks, It's for one hundred one that I aim.鈥� Said an amateur statistician Who wanted to help with this mission. 鈥淭wenty-one for the sample Standard deviation's ample: They examined one hundred and one Dalmatians that fetched a good sum. They counted each spot, Mark, freckle and dot And tallied up every one. Instead of one hundred one spots They averaged ninety six dots Can they muzzle Spreckles鈥� Obsession with freckles Based on all the dog data they've got?
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