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Chapter 8: Problem 1

For a binomial experiment with \(r\) successes out of \(n\) trials, what value do we use as a point estimate for the probability of success \(p\) on a single trial?

Short Answer

Expert verified
The point estimate for the probability of success \( p \) is the sample proportion, \( \hat{p} = \frac{r}{n} \).

Step by step solution

01

Understand the Problem

We need to find a point estimate for the probability of success \( p \) in a binomial experiment where we have \( r \) successes out of \( n \) trials. A point estimate is a single value that serves as an approximation for a population parameter.
02

Recall the Definition of a Point Estimate

In statistics, a common point estimate for a proportion is the sample proportion, which is calculated as the number of successes divided by the number of trials.
03

Calculate the Sample Proportion

Calculate the sample proportion, \( \hat{p} \), using the formula: \( \hat{p} = \frac{r}{n} \). Here, \( r \) is the number of successes and \( n \) is the total number of trials.
04

Interpret the Calculation

The resulting value of \( \hat{p} = \frac{r}{n} \) is our point estimate for the true probability of success \( p \) in the population. It is an estimate based on the sample data available.

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Key Concepts

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

Understanding a Binomial Experiment
A binomial experiment is a statistical experiment that features a fixed number of trials, where each trial has only two possible outcomes: success or failure. This type of experiment is quite common in various scenarios, such as flipping a coin or testing a batch of products.

There are four key properties of a binomial experiment:
  • Fixed Number of Trials: The experiment is conducted for a specific number of times, denoted by \( n \).
  • Binary Outcomes: Each trial results in either a success or a failure.
  • Constant Probability: The probability of success, denoted by \( p \), remains constant throughout all trials.
  • Independent Trials: The outcome of one trial does not affect the others.
By understanding these properties, you can apply the binomial experiment framework to a wide range of probability estimation problems, aiding in the prediction of outcomes where randomness is a factor.
What is a Point Estimate in Statistics?
A point estimate is a single numerical value used to approximate a population parameter. In the context of probability estimation, it helps us pinpoint an exact figure that serves as the most likely value for some unknown key aspect of a population.

There are several types of point estimates, depending on what parameter is being estimated:
  • For estimating a population mean, we use the sample mean.
  • For estimating a population variance, the sample variance is utilized.
  • When estimating a population proportion, as in binomial experiments, the sample proportion is the go-to estimator.
In essence, a point estimate gives us just one number to work with, representing our best guess based on sampled data from a larger set. This is an important tool in statistical analysis and decision-making.
The Role of Sample Proportion in Probability Estimation
The sample proportion is a simple yet powerful tool in probability estimation, especially when dealing with binomial experiments. It is represented by the symbol \( \hat{p} \) and is calculated using the formula: \( \hat{p} = \frac{r}{n} \), where \( r \) represents the number of successes, and \( n \) is the total number of trials.

Here's why the sample proportion is crucial:
  • Direct Estimation: The sample proportion directly reflects the observed frequency of success in the sample, making it an intuitive measure of the underlying probability of success \( p \).
  • Ease of Calculation: It is simple to compute, requiring just a division of two known measurements - successes and trials.
  • Foundation for Further Analysis: Once you have the sample proportion, it lays the groundwork for more complex inferencing, such as constructing confidence intervals or performing hypothesis tests.
Thus, the sample proportion acts as a bridge between raw data and meaningful probability estimates, facilitating deeper insights into stochastic processes and supporting empirical research.

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Most popular questions from this chapter

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

A random sample of medical files is used to estimate the proportion \(p\) of all people who have blood type \(B\). (a) If you have no preliminary estimate for \(p\), how many medical files should you include in a random sample in order to be \(85 \%\) sure that the point estimate \(\hat{p}\) will be within a distance of \(0.05\) from \(p ?\) (b) Answer part (a) if you use the preliminary estimate that about 8 out of 90 people have blood type B. (Reference: Manual of Laboratory and Diagnostic Tests, F. Fischbach.)

Lorraine was in a hurry when she computed a confidence interval for \(\mu .\) Because \(\sigma\) was not known, she used a Student's \(t\) distribution. However, she accidentally used degrees of freedom \(n\) instead of \(n-1 .\) Will her confidence interval be longer or shorter than one found using the correct degrees of freedom \(n-1 ?\) Explain.

In a combined study of northern pike, cutthroat trout, rainbow trout, and lake trout, it was found that 26 out of 855 fish died when caught and released using barbless hooks on flies or lures. All hooks were removed from the fish. (Source: A National Symposium on Catch and Release Fishing, Humboldt State University Press.) (a) Let \(p\) represent the proportion of all pike and trout that die (i.e., \(p\) is the mortality rate) when caught and released using barbless hooks. Find a point estimate for \(p\). (b) Find a \(99 \%\) confidence interval for \(p\), and give a brief explanation of the meaning of the interval. (c) Is the normal approximation to the binomial justified in this problem? Explain.

How hot is the air in the top (crown) of a hot air balloon? Information from Ballooning: The Complete Guide to Riding the Winds, by Wirth and Young (Random House), claims that the air in the crown should be an average of \(100^{\circ} \mathrm{C}\) for a balloon to be in a state of equilibrium. However, the temperature does not need to be exactly \(100^{\circ} \mathrm{C}\). What is a reasonable and safe range of temperatures? This range may vary with the size and (decorative) shape of the balloon. All balloons have a temperature gauge in the crown. Suppose that 56 readings (for a balloon in equilibrium) gave a mean temperature of \(\bar{x}=97^{\circ} \mathrm{C}\). For this balloon, \(\sigma \approx 17^{\circ} \mathrm{C}\). (a) Compute a \(95 \%\) confidence interval for the average temperature at which this balloon will be in a steady-state equilibrium. (b) If the average temperature in the crown of the balloon goes above the high end of your confidence interval, do you expect that the balloon will go up or down? Explain.
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