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Chapter 5: Problem 5

Critical Tbinking Consider a binomial distribution of 200 trials with expected value 80 and standard deviation of about \(6.9\). Use the criterion that it is unusual to have data values more than \(2.5\) standard deviations above the mean or \(2.5\) standard deviations below the mean to answer the following questions. (a) Would it be unusual to have more than 120 successes out of 200 trials? Explain. (b) Would it be unusual to have fewer than 40 successes out of 200 trials? Explain. (c) Would it be unusual to have from 70 to 90 successes out of 200 trials? Explain.

Short Answer

Expert verified
(a) Unusual; (b) Unusual; (c) Not unusual.

Step by step solution

01

Determine the Mean

The expected value (mean) of the binomial distribution is given as 80. This forms our central point for considering what constitutes an unusual event.
02

Calculate the Allowed Range

Use the criterion provided: data is unusual if it is more than 2.5 standard deviations from the mean. With standard deviation \(\sigma = 6.9\), compute the thresholds: \(\text{Lower bound} = 80 - 2.5 \times 6.9 \) and \(\text{Upper bound} = 80 + 2.5 \times 6.9\).
03

Lower Bound Calculation

Calculate the lower bound: \(80 - 2.5 \times 6.9 = 80 - 17.25 = 62.75\). Rounded, this means fewer than 63 successes is unusual.
04

Upper Bound Calculation

Calculate the upper bound: \(80 + 2.5 \times 6.9 = 80 + 17.25 = 97.25\). Rounded, this means more than 97 successes is unusual.
05

Check Case (a)

For part (a), 120 successes is proposed. This is more than the upper bound of 97. Hence, it is unusual to have more than 120 successes.
06

Check Case (b)

For part (b), 40 successes is proposed. This is less than the lower bound of 63. Hence, it is unusual to have fewer than 40 successes.
07

Check Case (c)

For part (c), evaluate the range from 70 to 90 successes. Both 70 and 90 fall within the calculated range of 63 to 97, so it is not unusual to have between 70 and 90 successes.

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Key Concepts

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

Binomial distribution
A binomial distribution is a type of probability distribution that arises from a type of experiment or process that has two possible outcomes. These outcomes are often referred to as "success" and "failure". Let's connect this to flipping a coin. Each flip represents one trial, and it can either come up heads (success) or tails (failure).
If we consider our exercise, we have 200 trials, which means we repeat our experiment 200 times.For a binomial distribution, we define two parameters:
  • Number of trials, commonly denoted as \( n \).
  • Probability of success on a single trial, denoted as \( p \).
In our exercise, the calculation of probable success rates draws from these basic principles of the binomial distribution, helping us realize the likelihoods of outcomes like 120 or 40 successes.
Standard deviation
Standard deviation is a measure that denotes the amount of variation or dispersion in a set of values. In simpler terms, it tells us how spread out the values are around the mean. The standard deviation in our problem is given as \(6.9\).
Here's how you can picture it:
  • A small standard deviation means data points are close to the mean.
  • A large standard deviation means data points are spread out over a larger range of values.
In the context of our exercise, knowing the standard deviation helps us set boundaries for what is usual or unusual by determining how far the outcomes can deviate from the mean. The criterion defined uses 2.5 times the standard deviation to classify results as unusual.
Mean
The mean is a statistical measure often referred to as the average. It provides us an overall idea of a dataset鈥檚 central value. In this binomial distribution problem, the mean is equal to the expected value, which is given as 80.
Calculating the mean in a binomial distribution is straightforward:
  • The formula is \( \mu = n \times p \).
For our scenario:
  • The mean value of 80 tells us that, out of 200 trials, you'd expect about 80 successes if each success is independent and randomly distributed.
This center point is essential in determining the regions of unusualness as it acts as the pivotal baseline around which the outcomes fluctuate.
Unusual events
Unusual events are outcomes that deviate from the expected results significantly. They are outside of the normal range defined by the mean and standard deviation in this context.
In our scenario:
  • An outcome is called unusual if it is more than \(2.5\) standard deviations away from the mean.
  • We calculate the lower and upper bounds using the formulas: \( \mu - 2.5 \sigma \) and \( \mu + 2.5 \sigma \).
    • This equates to boundaries of 63 and 97 for our example.
  • Results outside this range, such as more than 120 successes or fewer than 40 successes, are deemed unusual.
Using this criterion helps to identify results that might be statistically significant enough to question the initial hypothesis or assumptions.
Expected value
Expected value, essentially, is a prediction of the average outcome in a long series of trials. It is a key concept in understanding the behavior of random variables.
Calculation involves:
  • Multiplying each possible outcome by the probability of that outcome occurring, and summing these products.
  • In a binomial setting, it's simply \( n \times p \). Here, \( n \) is the number of trials and \( p \) the probability of success.
In our case, the expected value of 80 serves as the mean around which we base our judgments on what鈥檚 considered usual or unusual. As a quick reminder:
  • This value reflects what we anticipate on average across many trials.
  • It's important to remember that outcomes may not always hit this exact number, due to random chance and variability.
Having a solid grasp of expected value is invaluable for evaluating statistical experiments and interpreting results.

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Most popular questions from this chapter

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