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Chapter 7: Problem 2

In order to use a normal distribution to compute confidence intervals for \(p,\) what conditions on \(n p\) and \(n q\) need to be satisfied?

Short Answer

Expert verified
For a normal approximation, both \(np\) and \(nq\) must be \(\geq 5\).

Step by step solution

01

Understand the Parameters

First, identify that when dealing with a binomial distribution, where there are two outcomes (success and failure), the parameters are defined as follows: the probability of success is denoted as \(p\) and the probability of failure as \(q = 1 - p\). The number of trials is denoted as \(n\).
02

Recognize the Need for Normal Approximation

For the binomial distribution to approximate a normal distribution, it's crucial that both the number of expected successes \(np\) and the number of expected failures \(nq\) be large enough. This is to ensure that the distribution of sample proportions can be considered approximately normal.
03

Conditions for Normal Approximation

Statisticians commonly use the rule of thumb which states that both \(np\) and \(nq\) should be greater than or equal to 5. This rule helps ensure that the distribution of the sample proportions is close enough to a normal distribution.

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Key Concepts

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

Normal Distribution
A normal distribution, sometimes called a Gaussian distribution, is one of the most important concepts in statistics. This is because many things in the real world follow this kind of distribution. It has a characteristic bell-shaped curve when graphed. The curve is symmetric, meaning it looks the same on the left and right sides.

There are a couple of key features of a normal distribution:
  • Mean: The mean of the distribution is at the center of the curve.
  • Standard Deviation: This measures how spread out the numbers are from the mean. A small standard deviation means the data points are close to the mean; a large one means they are more spread out.
Normal distributions are really valuable because they allow us to make predictions about the data, calculate confidence intervals, and test hypotheses.
Binomial Distribution
The binomial distribution is a discrete probability distribution. It describes the number of successes in a fixed number of independent trials of a binary event. Here, binary means there are only two possible outcomes, such as success or failure.

Some important aspects of a binomial distribution include:
  • Fixed Number of Trials: The number of times the experiment is repeated is fixed.
  • Binary Outcomes: Each trial has only two outcomes - success or failure.
  • Constant Probability of Success: The probability of success remains the same for all trials.
One real-world example could be flipping a coin a set number of times and counting the number of heads you get. If you're dealing with a large number of trials, the binomial distribution starts to resemble a normal distribution.
Normal Approximation
Normal approximation to the binomial distribution is a method used to simplify calculations involving binomial probabilities. When a binomial distribution has a large number of trials, the computation can become complex. Fortunately, under certain conditions, it can be approximated by a normal distribution.

The conditions for using normal approximation are straightforward:
  • Rule of Thumb: Both the expected number of successes (np) and the expected number of failures (nq) should be 5 or greater.
  • Larger Sample Sizes: The larger the sample size, the better the approximation.
This method helps because it allows the use of normal distribution formulas and tables, making computations easier and more efficient.
Probability of Success
Probability of success, denoted as \( p \), is a fundamental component of the binomial distribution. It represents the chance or likelihood that a single trial of an experiment will result in success.

Key features of probability of success (\( p \)) include:
  • Range: \( p \) is always between 0 and 1, inclusive. A\( p \) of 0 means there is no chance of success, while a \( p \) of 1 means success is guaranteed.
  • Complement: The probability of failure is \( q = 1 - p \).
  • Stability: In a binomial distribution, the probability of success remains constant across all trials.
Understanding \( p \) and its complement \( q \) is essential for calculating probabilities and making predictions in binomial processes.

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Most popular questions from this chapter

Diagnostic Tests: Total Calcium Over the past several months, an adult patient has been treated for tetany (severe muscle spasms). This condition is associated with an average total calcium level below 6 mg/dl (Reference: Manual of Laboratory and Diagnostic Tests by F. Fischbach). Recently, the patient's total calcium tests gave the following readings (in mg/dl). $$ \begin{array}{ccccccc} 9.3 & 8.8 & 10.1 & 8.9 & 9.4 & 9.8 & 10.0 \\ 9.9 & 11.2 & 12.1 & & & & \end{array} $$ (a) Use a calculator to verify that \(\bar{x}=9.95\) and \(s \approx 1.02\) (b) Find a \(99.9 \%\) confidence interval for the population mean of total calcium in this patient's blood. (c) Interpretation Based on your results in part (b), does it seem that this patient still has a calcium deficiency? Explain.

Jobs and productivity! How do retail stores rate? One way to answer this question is to examine annual profits per employee. The following data give annual profits per employee (in units of 1 thousand dollars per employee) for companies in retail sales. (See reference in Problem \(23 .\) ) Companies such as Gap, Nordstrom, Dillards, JCPenney, Sears, Wal-Mart, Office Depot, and Toys " \(\mathrm{A}\) " Us are included. Assume \(\sigma \approx 3.8\) thousand dollars. $$\begin{array}{rrrrrrrrrrrr}4.4 & 6.5 & 4.2 & 8.9 & 8.7 & 8.1 & 6.1 & 6.0 & 2.6 & 2.9 & 8.1 & -1 . \\\11.9 & 8.2 & 6.4 & 4.7 & 5.5 & 4.8 & 3.0 & 4.3 & -6.0 & 1.5 & 2.9 & 4 . \\ -1.7 & 9.4 & 5.5 & 5.8 & 4.7 & 6.2 & 15.0 & 4.1 & 3.7 & 5.1 & 4.2 &\end{array}$$ (a) Use a calculator or appropriate computer software to verify that, for the preceding data, \(\bar{x} \approx 5.1 .\) (b) Let us say that the preceding data are representative of the entire sector of retail sales companies. Find an \(80 \%\) confidence interval for \(\mu,\) the average annual profit per employee for retail sales. (c) Interpretation Let us say that you are the manager of a retail store with a large number of employees. Suppose the annual profits per employee are less than 3 thousand dollars per employee. Do you think this might be low compared with other retail stores? Explain by referring to the confidence interval you computed in part (b). (d) Interpretation Suppose the annual profits are more than 6.5 thousand dollars per employee. As store manager, would you feel somewhat better? Explain by referring to the confidence interval you computed in part (b). (e) Repeat parts \((b),(c),\) and (d) for a \(95 \%\) confidence interval.

Student's \(t\) distributions are symmetric about a value of \(t .\) What is that \(t\) value?

What percentage of your campus student body is female? Let \(p\) be the proportion of women students on your campus. (a) If no preliminary study is made to estimate \(p,\) how large a sample is needed to be \(99 \%\) sure that a point estimate \(\hat{p}\) will be within a distance of 0.05 from \(p ?\) (b) The Statistical Abstract of the United States, 1 12th edition, indicates that approximately \(54 \%\) of college students are female. Answer part (a) using this estimate for \(p\).

How hard is it to reach a businessperson by phone? Let \(p\) be the proportion of calls to business people for which the caller reaches the person being called on the first try. (a) If you have no preliminary estimate for \(p,\) how many business phone calls should you include in a random sample to be \(80 \%\) sure that the point estimate \(\hat{p}\) will be within a distance of 0.03 from \(p ?\) (b) The Book of Odds by Shook and Shook (Signet) reports that businesspeople can be reached by a single phone call approximately \(17 \%\) of the time. Using this (national) estimate for \(p,\) answer part (a).
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