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Chapter 8: Problem 2

Use the following information to answer the next five exercises: The standard deviation of the weights of elephants is known to be approximately 15 pounds. We wish to construct a 95% confidence interval for the mean weight of newborn elephant calves. Fifty newborn elephants are weighed. The sample mean is 244 pounds. The sample standard deviation is 11 pounds. In words, define the random variables \(X\) and \(\overline{X}\) .

Short Answer

Expert verified
\(X\) is the weight of a single newborn elephant, and \(\overline{X}\) is the average weight of 50 newborn elephants.

Step by step solution

01

Understand the Random Variable X

The random variable \(X\) represents the weight of a single newborn elephant calf from the population of newborn elephant calves. Since we are sampling a group of newborn elephants, \(X\) is considered to represent individual data points within this population.
02

Define the Sample Mean \( \overline{X} \)

The symbol \( \overline{X} \) represents the sample mean weight of the 50 newborn elephant calves we have in our study. It is calculated by averaging the weights of these 50 elephants. In this context, it serves as an estimate of the true population mean weight of newborn elephant calves.

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Key Concepts

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

Random Variables
Random variables are a crucial concept in statistics and probability, representing numerical outcomes of random phenomena. In this exercise, the random variable \(X\) is defined as the weight of a single newborn elephant calf from the entire population of newborn calves collected for our study. Each calf represents a possible outcome, making \(X\) a representation of individual data points.

Random variables can be:
  • Discrete: possible outcomes are distinct and countable.
  • Continuous: possible outcomes form a continuum and may take any value within a range.
In our context, \(X\) is a continuous random variable since the weight can theoretically take any real value within the range of possible weights for elephant calves.
Sample Mean
The sample mean, denoted by \(\overline{X}\), serves as a vital estimator in statistical analysis. In our case, \(\overline{X}\) is the average weight of the 50 newborn elephant calves sampled in the study. It's calculated by summing their weights and dividing by the number of elephants, which provides us with a value that approximates the population mean.

The sample mean is utilized to:
  • Estimate the population mean.
  • Form the basis for constructing confidence intervals.
  • Analyze the central tendency of the sample data.
By averaging the weights of the sample group, \(\overline{X} = 244\) pounds gives an indication of where the center of the distribution of weights lies for the chosen sample.
Population Mean
The population mean, often symbolized as \(\mu\), is the average of all data points in a population. It represents the central value of the entire group, unlike the sample mean which is calculated from just a part of the population.

In statistics, the population mean acts as a:
  • Benchmark for comparing individual or group data.
  • Key parameter in hypothesis testing.
  • Measure to determine the accuracy of the sample mean as an estimator.
In our exercise, though we do not know the actual population mean of newborn elephant calves, we aim to estimate \(\mu\) by constructing a confidence interval using the sample data. This involves statistical techniques to infer the possible range of the true mean based on observed data.
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. It quantifies how much individual data points deviate on average from the mean of the data set.

There are two key types of standard deviation:
  • Sample Standard Deviation: Calculated for a sample, helps in understanding data spread within a sample.
  • Population Standard Deviation: Applicable to the entire population, but this is often unknown and estimated from the sample data.
In our problem, the standard deviation for the weights of newborn elephants is known to be approximately 15 pounds, considered to be the population standard deviation. On the other hand, the sample standard deviation of the 50 elephants surveyed is 11 pounds, which provides insight into the consistency of the weights within this sample group.

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Most popular questions from this chapter

Forbes magazine published data on the best small firms in 2012 . These were firms that had been publicly traded for at least a year, have a stock price of at least \(\$ 5\) per share, and have reported annual revenue between \(\$ 5\) million and \(\$ 1\) billion. The Table 8.13 shows the ages of the corporate \(\mathrm{CEOs}\) for a random sample of these firms. $$\begin{array}{|c|c|c|c|c|}\hline 48 & {58} & {51} & {61} & {56} \\ \hline 59 & {74} & {63} & {53} & {50} \\ \hline 59 & {60} & {60} & {57} & {46} \\\ \hline 55 & {63} & {57} & {47} & {55} \\ \hline 57 & {43} & {61} & {62} & {49} \\ \hline 67 & {67} & {55} & {55} & {49} \\ \hline\end{array}$$ Use this sample data to construct a 90% confidence interval for the mean age of CEO鈥檚 for these top small firms. Use the Student's t-distribution.

Use the following information to answer the next three exercises: According to a Field Poll, 79% of California adults (actual results are 400 out of 506 surveyed) feel that education and our schools is one of the top issues facing California. We wish to construct a 90% confidence interval for the true proportion of California adults who feel that education and the schools is one of the top issues facing California. A 90\(\%\) confidence interval for the population proportion is _____. a. \((0.761,0.820)\) b. \((0.125,0.188)\) c. \((0.755,0.826)\) d. \((0.130,0.183)\)

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Use the following information to answer the next five exercises. A hospital is trying to cut down on emergency room wait times. It is interested in the amount of time patients must wait before being called back to be examined. An investigation committee randomly surveyed 70 patients. The sample mean was 1.5 hours with a sample standard deviation of 0.5 hours. Identify the following: a. \(x=\) b. \(s_{x}=\) C. \(n=\) d. \(n-1=\)
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