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Chapter 8: Problem 7

Use the following information to answer the next seven exercises: The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The Bureau surveys 200 people. The sample mean is 8.2 minutes. There is a known standard deviation of 2.2 minutes. The population distribution is assumed to be normal. In words, define the random variables \(X\) and \(\overline{X}\) .

Short Answer

Expert verified
\(X\) is the time for one person; \(\overline{X}\) is the average time for 200 people.

Step by step solution

01

Defining Random Variable X

The random variable \(X\) represents the individual time taken by a person to complete the short form. Each value of \(X\) is a single observation from the population, measured in minutes.
02

Defining Random Variable X-bar

The random variable \(\overline{X}\) is the sample mean time taken by a group of 200 people to complete the short form. It is the average of all individual times \(X\) obtained from the sample.

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Key Concepts

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

Random Variables
The concept of random variables is fundamental in statistics as it helps to describe outcomes in a probabilistic framework. A random variable is essentially a numeric representation of the outcomes of a random process, allowing us to quantify and analyze data. In this exercise, we have two types of random variables:
  • **Random Variable X:** This represents the time each individual takes to complete the survey. Every participant may have a different timing, and these variations are captured by the random variable \(X\). It helps us model the individualized completion times, making complex data manageable.
  • **Random Variable X-bar:** Denoted as \(\overline{X}\), this is the sample mean, representing the average time taken by the surveyed participants. This random variable helps condense all individual observations into a single, meaningful statistic, which is crucial for further analysis.
Understanding these variables makes it easier to create models and draw conclusions about larger populations based on sample data.
Sample Mean
The sample mean, represented symbolically as \(\overline{X}\), is an essential statistic in data analysis. It serves as an estimate of the true population mean, offering insights into the central tendency of a dataset. In our case, \[\overline{X} = 8.2 \ ext{minutes}\]This value represents the average time all 200 surveyed individuals took to complete the form. Here are a few reasons why the sample mean is significant:
  • **Central Tendency:** The sample mean gives a single value summarizing the entire data set, helping to understand the general behavior of the data without analyzing each individual point.
  • **Estimating Population Mean:** As a representative of the population, the sample mean extrapolates the overall trend seen in the sample to predict the behavior of the whole population.
  • **Simplicity and Efficiency:** Calculating a mean is straightforward, making it a reliable tool for initial descriptive statistics.
Moreover, when combined with other statistics, the sample mean can help build confidence intervals and conduct hypothesis tests, which are pivotal in making informed decisions based on data.
Standard Deviation
Standard deviation is a crucial measure in statistics, indicating how much individual data points in a dataset deviate from the mean. It provides insight into the data's variability or spread. For our dataset, the known standard deviation is \[\sigma = 2.2 \, \text{minutes}\]Here's why understanding standard deviation matters:
  • **Variation Insight:** A lower standard deviation implies that the data points are close to the mean, indicating consistency, while a higher standard deviation suggests greater variability.
  • **Interpreting Data Spread:** It helps to quantify the extent of variability or diversity in the dataset, which is not visible with the mean alone.
  • **Comparison Tool:** Enables comparison between different datasets by providing a common metric for variability, especially when comparing samples of different sizes or scales.
  • **Statistical Inferences:** Standard deviation is foundational for calculating other statistics, such as variance and z-scores, and is critical for normal distribution assumptions.
Thus, standard deviation not only aids in understanding the nature of the dataset but is also integral to statistical testing, making it a powerful tool for comprehensive data analysis.

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Most popular questions from this chapter

Use the following information to answer the next 13 exercises: The data in Table 8.10 are the result of a random survey of 39 national flags (with replacement between picks) from various countries. We are interested in finding a confidence interval for the true mean number of colors on a national flag. Let X = the number of colors on a national flag. $$\begin{array}{|c|c|}\hline X & {\text { Freq }} \\ \hline 1 & {1} \\\ \hline 2 & {7} \\ \hline 3 & {78} \\ \hline 4 & {7} \\ \hline 5 & {6} \\\ \hline\end{array}$$ What is \(\overline{x}\) estimating?

Use the following information to answer the next five exercises: of \(1,050\) randomly selected adults, 360 identified themselves as manual laborers, 280 identified themselves as non-manual wage earners, 250 identified themselves as mid-level managers, and 160 identified themselves as executives. In the survey, 82% of manual laborers preferred trucks, 62% of non-manual wage earners preferred trucks, 54% of mid-level managers preferred trucks, and 26% of executives preferred trucks. Suppose we want to lower the sampling error. What is one way to accomplish that?

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. Identify the following: a. \(\overline{x}=\) b. \(s_{x}=\) C. \(n=\) d. \(n-1=\)

Use the following information to answer the next 16 exercises: The Ice Chalet offers dozens of different beginning ice-skating classes. All of the class names are put into a bucket. The 5 P.M., Monday night, ages 8 to 12, beginning ice-skating class was picked. In that class were 64 girls and 16 boys. Suppose that we are interested in the true proportion of girls, ages 8 to 12, in all beginning ice-skating classes at the Ice Chalet. Assume that the children in the selected class are a random sample of the population. Calculate the following: a. \(x=\) _____ b. \(n=\) _____ c. \(p^{\prime}=\) _____

Use the following information to answer the next 16 exercises: The Ice Chalet offers dozens of different beginning ice-skating classes. All of the class names are put into a bucket. The 5 P.M., Monday night, ages 8 to 12, beginning ice-skating class was picked. In that class were 64 girls and 16 boys. Suppose that we are interested in the true proportion of girls, ages 8 to 12, in all beginning ice-skating classes at the Ice Chalet. Assume that the children in the selected class are a random sample of the population. Using the same \(p^{\prime}\) and \(n=80,\) how would the error bound change if the confidence level were increased to 98\(\% ?\) Why?
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