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Chapter 8: Problem 27

Use the following information to answer the next 14 exercises: The mean age for all Foothill College students for a recent Fall term was 33.2. The population standard deviation has been pretty consistent at 15. Suppose that twenty-five Winter students were randomly selected. The mean age for the sample was 30.4. We are interested in the true mean age for Winter Foothill College students. Let X = the age of a Winter Foothill College student. What is \(\overline{x}\) estimating?

Short Answer

Expert verified
   ame{t}he tru ame {e}  ame age of ame{W}inter st ame {ud}ents.

Step by step solution

01

Understanding the Given Values

Here, we have the mean age for all Foothill College students as 33.2, population standard deviation as 15, and a sample mean age of 30.4 for 25 randomly selected Winter students. We aim to estimate the true mean age of all Winter students.
02

Identifying the Population Parameter

The exercise involves using a sample (25 Winter students) to make inferences about a larger population (all Winter students). The sample mean ( 30.4) estimates the population mean ( X ) for Winter students.

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Key Concepts

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

Mean Estimation
When dealing with statistics, mean estimation is a core concept. It involves using sample data to make an educated guess about the population mean. The mean, often represented by the symbol \( \mu \) for populations or \( \overline{x} \) for samples, is the average value among a set of numbers. In the context of the exercise:
  • The sample mean \( \overline{x} \) is calculated as the average age of the 25 sampled Winter students, which is 30.4 years.
  • This calculated mean is used to provide an estimate of the true mean age of all Winter students at Foothill College.
By estimating the mean, we make inferences about the population using the sample data. This process is significant because it allows us to draw conclusions about a population without needing data from every single individual.
Population Parameter
The concept of a population parameter is essential in statistical inference. A parameter is a value that describes a characteristic of an entire population.In this exercise:
  • The parameter of interest is the population mean age of all Winter students at Foothill College, often represented as \( \mu \).
Population parameters are typically unknown because it's impractical or impossible to measure an entire population directly. Hence, we rely on sample statistics (like \( \overline{x} \), the sample mean) to estimate these parameters. Understanding the population parameter helps clarify what you're ultimately trying to learn or predict from the sample data.
Sample Mean
The sample mean is a statistic that provides useful insights into a larger population. It is calculated as the average of the sampled data and is denoted by \( \overline{x} \).Here's how it works:
  • In our exercise, the sample mean of 30.4 was calculated by averaging the ages of the 25 randomly selected Winter students.
  • \( \overline{x} = \frac{\sum X_i}{n} \), where \( \sum X_i \) is the sum of all sampled ages and \( n \) is the number of students sampled.
The sample mean serves as an estimate of the population mean. It's a central part of the statistical toolkit because it provides a simple, tangible piece of evidence about the population from which the sample is drawn.

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Most popular questions from this chapter

Use the following information to answer the next ten exercises: A sample of 20 heads of lettuce was selected. Assume that the population distribution of head weight is normal. The weight of each head of lettuce was then recorded. The mean weight was 2.2 pounds with a standard deviation of 0.1 pounds. The population standard deviation is known to be 0.2 pounds. In words, define the random variable \(X .\)

Use the following information to answer the next two exercises: A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16 oz. serving size. The sample mean is 13.30 with a sample standard deviation of 1.55. Assume the underlying population is normally distributed. Find the 95\(\%\) Confidence Interval for the true population mean for the amount of soda served. a. \((12.42,14.18)\) b. \((12.32,14.29)\) C. \((12.50,14.10)\) d. Impossible to determine

Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. We wish to construct a 95% confidence interval for the mean height of male Swedes. Forty-eight male Swedes are surveyed. The sample mean is 71 inches. The sample standard deviation is 2.8 inches. a. i. \(\overline{x}=\) _____ ii. \(\sigma=\) _____ iii. \(n=\) _____ b. In words, define the random variables \(X\) and \(\overline{X}\) . c. Which distribution should you use for this problem? Explain your choice. d. Construct a 95\(\%\) confidence interval for the population mean height of male Swedes. i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound. e. What will happen to the level of confidence obtained if \(1,000\) male Swedes are surveyed instead of 48\(?\) Why?

Use the following information to answer the next six exercises: One hundred eight Americans were surveyed to determine the number of hours they spend watching television each month. It was revealed that they watched an average of 151 hours each month with a standard deviation of 32 hours. Assume that the underlying population distribution is normal. Define the random variable \(X\) in words.

A national survey of 1,000 adults was conducted on May 13,2013 by Rasmussen Reports. It concluded with \(95 \%\) confidence that \(49 \%\) to \(55 \%\) of Americans believe that big-time college sports programs corrupt the process of higher education. a. Find the point estimate and the error bound for this confidence interval. b. Can we (with \(95 \%\) confidence) conclude that more than half of all American adults believe this? c. Use the point estimate from part a and \(n=1,000\) to calculate a \(75 \%\) confidence interval for the proportion of American adults that believe that major college sports programs corrupt higher education. d. Can we (with \(75 \%\) confidence) conclude that at least half of all American adults believe this?
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