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Chapter 6: Problem 22

A manufacturer of window frames knows from long experience that 5 percent of the production will have some type of minor defect that will require an adjustment. What is the probability that in a sample of 20 window frames: a. None will need adjustment? b. At least one will need adjustment? c. More than two will need adjustment?

Short Answer

Expert verified
a. 0.3585; b. 0.6415; c. 0.0756.

Step by step solution

01

Understanding the Problem

We are given that each window frame has a 5% probability of having a defect, which means 5% will need adjustment. We use a binomial distribution where the number of trials (frames) is 20 and the probability of success (defect) is 0.05.
02

Define the Binomial Distribution

In a binomial distribution, the probability of getting exactly k successes in n trials is given by \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \). Here, \( n = 20 \), \( p = 0.05 \).
03

Calculate Probability for Part (a): None will need adjustment

For none needing adjustment, \( k = 0 \). Use the formula: \[ P(X = 0) = \binom{20}{0} (0.05)^0 (0.95)^{20} \]. This simplifies to \( (0.95)^{20} \). Compute this.
04

Calculate Probability for Part (b): At least one will need adjustment

To find the probability that at least one frame needs adjustment, use the complement: 1 minus the probability that none need adjustment. That is, \( P(X \geq 1) = 1 - P(X = 0) \).
05

Calculate Probability for Part (c): More than two will need adjustment

For more than two needing adjustment, calculate \( P(X > 2) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)] \). First, calculate \( P(X = 1) \) using \[ \binom{20}{1} (0.05)^1 (0.95)^{19} \] and \( P(X = 2) \) using \[ \binom{20}{2} (0.05)^2 (0.95)^{18} \]. Sum these probabilities and subtract from 1.
06

Final Calculations

Using a calculator: - \( P(X = 0) = (0.95)^{20} \approx 0.3585 \).- \( P(X \geq 1) = 1 - 0.3585 = 0.6415 \).- \( P(X = 1) \approx 0.3774 \) and \( P(X = 2) \approx 0.1885 \).- \( P(X > 2) = 1 - (0.3585 + 0.3774 + 0.1885) \approx 0.0756 \).

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Key Concepts

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

Understanding Probability
Probability is the core metric used to determine the likelihood of an event happening. In simple terms, it describes how likely something is to occur. For example, if we say there is a 5% probability that a window frame will have a defect, it means out of every 100 window frames, around 5 might have defects on average.

Probability is always expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 signifies an event that is certain to happen. The probability of all possible outcomes of a trial adds up to 1.

To find the probability of none of the frames needing an adjustment, we use the binomial formula:
  • First, identify the number of trials and the probability of success. Here, the number of trials (n) is 20, and the probability of a defect (success) p is 0.05.
  • For none needing adjustment, we set the number of successful events (k) to 0.
  • Then apply the formula for the probability of k successes: \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( \binom{n}{k} \) is the binomial coefficient.
  • This result tells us the probability of none of the samples having a defect.
The Role of Statistics
Statistics plays a crucial role when analyzing probabilities, as it allows us to gather, review, and interpret data to make inferences about population parameters based on samples. For instance, using the probability of defects, one can predict the performance of the manufacturing process. This is referred to as statistical inference.

The tools of statistics allow us to take a sample (like the 20 window frames) and make predictions about the larger population of all produced window frames. A binomial distribution in statistics is an example of using statistical methods to predict outcomes over several observations.

In this case:
  • We calculate the chance that none of the frames needed adjustments.
  • Then determine the likelihood of at least one needing adjustment, which is 1 minus the probability of none needing adjustment.
  • The calculations demonstrate how statistical methods simplify understanding outcomes in production scenarios.
Principles of Statistical Analysis
Statistical analysis involves the process of collecting and exploring data to uncover patterns and trends. In our example, we engage in statistical analysis by calculating probabilities for different outcomes, using a binomial distribution model.

When we examine the probability of more than two window frames needing adjustment, we perform several key steps:
  • First, we need to calculate the probability of exactly 0, 1, and 2 frames needing adjustment.
  • This involves using the binomial probability formula for each case and adding the probabilities for 0, 1, and 2 successful events.
  • Finally, subtract this sum from 1 to find the probability of more than two frames needing adjustment.
  • This type of analysis helps us understand the production quality control and areas that might need attention to reduce defects.
By engaging in this type of analysis, manufacturers can refine processes and address potential issues early, ensuring higher quality and fewer defects in future production runs.

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Most popular questions from this chapter

Croissant Bakery, Inc. offers special decorated cakes for birthdays, weddings, and other occasions. They also have regular cakes available in their bakery. The following table gives the total number of cakes sold per day and the corresponding probability. Compute the mean, variance, and standard deviation of the number of cakes sold per day. $$\begin{array}{|cc|}\hline \text { Number of Cakes } & \\\\\text { Sold in a Day } & \text { Probability } \\\\\hline 12 & .25 \\\13 & .40 \\\14 & .25 \\\15 & .10 \\\\\hline\end{array}$$

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