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Chapter 6: Problem 8

How can relative frequencies be used to help us estimate probabilities occurring in sampling distributions?

Short Answer

Expert verified
Relative frequencies can estimate probabilities in sampling distributions by approaching true probabilities through repeated sampling.

Step by step solution

01

Understanding Relative Frequency

Relative frequency is the fraction or proportion of the number of times an outcome occurs relative to the total number of trials or observations. It is calculated using the formula: \( \text{Relative Frequency} = \frac{\text{Frequency of an outcome}}{\text{Total number of trials}} \).
02

Connecting Relative Frequency to Probability

In probability theory, the probability of an event is defined as the measure of the likelihood that the event will occur. When empirical data is used, the relative frequency approaches the true probability as the number of trials increases. This is due to the Law of Large Numbers, which states that as the number of experiments increases, the relative frequency of an event approaches the theoretical probability.
03

Applying Relative Frequency in Sampling Distributions

A sampling distribution represents the probability distribution of a statistic (like the mean) obtained from a large number of samples drawn from a specific population. By calculating the relative frequencies of these samples, we can estimate the probabilities of different outcomes, thus building an empirical probability distribution that predicts the behavior of the statistic under repeated sampling.
04

Example and Interpretation

Suppose we want to estimate the probability that a sample mean falls within a certain range. By taking repeated samples from the population and calculating the sample means, we can determine the relative frequency of these means falling within the desired range. As the number of samples increases, this relative frequency provides an accurate estimate of the probability for the sample means in that range.

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Key Concepts

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

Relative Frequency
Relative frequency is a concept used to understand how often an event happens compared to the total number of trials or observations. It is a simple calculation: take the number of times the event of interest occurs, and divide it by the total number of trials. For instance, if you flip a coin 100 times and it lands heads up 55 times, the relative frequency of getting heads is 55/100 or 0.55.
  • Easy formula: \( \text{Relative Frequency} = \frac{\text{Frequency of an outcome}}{\text{Total number of trials}} \)
  • It's fractional, meaning it ranges from 0 to 1.
This measurement is essential because it provides a way to understand empirical data, or data that comes from experiments and observations. As more trials are conducted, this relative frequency becomes a more reliable indicator of what will happen in future trials as well.
Probability Estimation
Estimating probability using relative frequency involves observing the outcome of numerous trials and using the frequency with which specific outcomes occur as an estimate for probability. This practical approach allows us to make predictions about future events based on past data.
When we perform experiments, such as rolling a die repeatedly, the data we collect becomes our empirical evidence. Take note:
  • Probability is a fundamental concept in statistics, representing the chance of a particular outcome.
  • If you roll a die 600 times and it lands on three 100 times, the estimated probability of rolling a three is approximately 0.167 based on your data.
As the number of trials increases, our estimation becomes more accurate. This is the beginning of linking probability and relative frequency, helping us to see the likelihood of an outcome through observed patterns.
Law of Large Numbers
The Law of Large Numbers is a basic principle in probability and statistics that connects relative frequency and theoretical probability. It states that as the number of experiments or trials in a random process increases, the relative frequency of an event's occurrence becomes closer to the theoretical, or expected, probability.
  • This law applies to a variety of contexts, like flipping a coin or drawing cards.
  • For instance, flipping a fair coin should theoretically yield heads 50% of the time. With many coin flips, say thousands, the observed frequency will tend to converge near this theoretical probability of 0.5.
In essence, this means gathering more data helps improve the accuracy of your predictions, grounding them in statistical reality. Understanding this law is crucial for making valid estimations and predictions in many fields of study and practical applications.

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Most popular questions from this chapter

Find the \(z\) value described and sketch the area described. Find the \(z\) value such that \(98 \%\) of the standard normal curve lies between \(-z\) and \(z\).

Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(-0.82 \leq z \leq 0) $$

Let \(z\) be a random variable with a standard normal distribution. Find the indicated probability, and shade the corresponding area under the standard normal curve. $$ P(-0.73 \leq z \leq 3.12) $$

Let \(x\) be a random variable that represents checkout time (time spent in the actual checkout process) in minutes in the express lane of a large grocery. Based on a consumer survey, the mean of the \(x\) distribution is about \(\mu=2.7\) minutes, with standard deviation \(\sigma=0.6\) minute. Assume that the express lane always has customers waiting to be checked out and that the distribution of \(x\) values is more or less symmetrical and mound-shaped. What is the probability that the total checkout time for the next 30 customers is less than 90 minutes? Let us solve this problem in steps. (a) Let \(x_{i}\) (for \(\left.i=1,2,3, \ldots, 30\right)\) represent the checkout time for each customer. For example, \(x_{1}\) is the checkout time for the first customer, \(x_{2}\) is the checkout time for the second customer, and so forth. Each \(x_{i}\) has mean \(\mu=2.7\) minutes and standard deviation \(\sigma=0.6\) minute. Let \(w=x_{1}+x_{2}+\cdots+x_{30}\) Explain why the problem is asking us to compute the probability that \(w\) is less than 90 . (b) Use a little algebra and explain why \(w<90\) is mathematically equivalent to \(w / 30<3 .\) Since \(w\) is the total of the \(30 x\) values, then \(w / 30=\bar{x}\). Therefore, the statement \(\bar{x}<3\) is equivalent to the statement \(w<90\). From this we conclude that the probabilities \(P(\bar{x}<3)\) and \(P(w<90)\) are equal. (c) What does the central limit theorem say about the probability distribution of \(\bar{x} ?\) Is it approximately normal? What are the mean and standard deviation of the \(\bar{x}\) distribution? (d) Use the result of part (c) to compute \(P(\bar{x}<3)\). What does this result tell you about \(P(w<90)\) ?

Let \(x\) be a random variable that represents the level of glucose in the blood (milligrams per deciliter of blood) after a 12 -hour fast. Assume that for people under 50 years old, \(x\) has a distribution that is approximately normal, with mean \(\mu=85\) and estimated standard deviation \(\sigma=25\) (based on information from Diagnostic Tests with Nursing Applications, edited by S. Loeb, Springhouse). A test result \(x<40\) is an indication of severe excess insulin, and medication is usually prescribed. (a) What is the probability that, on a single test, \(x<40\) ? (b) Suppose a doctor uses the average \(\bar{x}\) for two tests taken about a week apart. What can we say about the probability distribution of \(\bar{x}\) ? Hint: See Theorem 6.1. What is the probability that \(\bar{x}<40\) ? (c) Repeat part (b) for \(n=3\) tests taken a week apart. (d) Repeat part (b) for \(n=5\) tests taken a week apart. (e) Compare your answers to parts (a), (b), (c), and (d). Did the probabilities decrease as \(n\) increased? Explain what this might imply if you were a doctor or a nurse. If a patient had a test result of \(\bar{x}<40\) based on five tests, explain why either you are looking at an extremely rare event or (more likely) the person has a case of excess insulin.
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