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Chapter 14: Problem 8

An automatic filling machine in a factory fills bottles of ketchup with a mean of 16.1 oz and a standard deviation of 0.05 oz with a distribution that can be well modeled by a Normal model. What is the probability that your bottle of ketchup contains less than 16 oz?

Short Answer

Expert verified
The probability that your bottle of ketchup contains less than 16 oz is 0.0228 or 2.28%.

Step by step solution

01

Identify the parameters of the Normal Distribution

The problem tells that weight of ketchup filled by the machine follows a Normal Distribution. The mean (\( \mu \)) is 16.1 oz and the standard deviation (\( \sigma \)) is 0.05 oz.
02

Calculate the z-score

The Z score is a measure of how many standard deviations an element is from the mean. To find probability that a bottle has less than 16 oz of ketchup, we need to find the z-score of 16. The formula of z-score is \( z = \frac{x - \mu}{\sigma} \), where x is the value that we're finding the probability for. In this case, x = 16, \( \mu \) = 16.1, and \( \sigma \) = 0.05. Plugging in these values, we get \( z = \frac{16 - 16.1}{0.05} = -2 \). This means that 16 oz is 2 standard deviations below the mean.
03

Find the probability corresponding to the z-score

Now, in order to find the probability of having less than 16 oz, look up the probability corresponding to z = -2 on the standard normal distribution table (or use a technology tool). The table value at z = -2 is 0.0228 or 2.28%.

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Key Concepts

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

Understanding the Z-Score
The **z-score** is an important concept in statistics, especially when working with normal distributions. It tells us how many standard deviations a specific value is from the mean. This helps us determine how unusual or typical a particular data point is in a distribution.
To calculate the z-score, you use the formula: \[ z = \frac{x - \mu}{\sigma} \] where:
  • \( x \) is the value in question.
  • \( \mu \) is the mean of the distribution.
  • \( \sigma \) is the standard deviation.

This calculation places the value on the standard normal distribution, which is a special kind of normal distribution with a mean of 0 and a standard deviation of 1. Once you have the z-score, you can find out what percentage of data falls below that value.
The Role of Standard Deviation
**Standard deviation** is a measure of how spread out the values in a data set are. It's a key concept in understanding a normal distribution. When you have a smaller standard deviation, most values are closer to the mean. Conversely, a larger standard deviation indicates more variability in the dataset.
Calculating standard deviation involves several steps:
  • First, find the mean (average).
  • Subtract the mean from each data point and square the result.
  • Find the average of these squared differences.
  • Take the square root of this average, and you have the standard deviation.

In the context of our bottle-filling example, the standard deviation helps us understand how much individual bottle fills deviate from the average amount of 16.1 oz.
Probability and Its Calculation
**Probability** quantifies the likelihood of an event happening and is crucial in statistics. When dealing with a normal distribution, probability helps us find the odds of a particular outcome, like a bottle containing less than 16 oz.
To calculate this probability, we first determine the z-score. Using the z-score, we refer to the standard normal distribution table or use software tools to find the corresponding probability value.
  • The z-score tells us how many standard deviations a value is from the mean.
  • The probability found in the table reflects how much of the distribution falls below this z-score.

In our case, a z-score of -2 corresponds to a probability of 0.0228. This means there's a 2.28% chance a bottle will have less than 16 oz of ketchup.
The Importance of the Mean
The **mean** is one of the most fundamental concepts in statistics. It represents the average value of a dataset. In a normally distributed dataset, the mean is the central point where the highest peak of the bell curve occurs.
To find the mean, you sum up all the data points and divide by the number of points.
  • For our ketchup bottles, a mean of 16.1 oz tells us that on average, each bottle is filled with this amount.
  • A mean is useful for comparing individual data points and understanding their deviation in context.

The mean also plays a critical role in calculating the z-score and analyzing the spread of a distribution, as it's a reference point for measuring variability.

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Most popular questions from this chapter

The amount of cereal that can be poured into a small bowl varies with a mean of 1.5 ounces and a standard deviation of 0.3 ounces. A large bowl holds a mean of 2.5 ounces with a standard deviation of 0.4 ounces. You open a new box of cereal and pour one large and one small bowl. a. How much more cereal do you expect to be in the large bowl? b. What's the standard deviation of this difference? c. If the difference follows a Normal model, what's the probability the small bowl contains more cereal than the large one? d. What are the mean and standard deviation of the total amount of cereal in the two bowls? e. If the total follows a Normal model, what's the probability you poured out more than 4.5 ounces of cereal in the two bowls together? f. The amount of cereal the manufacturer puts in the boxes is a random variable with a mean of 16.3 ounces and a standard deviation of 0.2 ounces. Find the expected amount of cereal left in the box and the standard deviation.

Pick a card, any card You draw a card from a deck. If you get a red card, you win nothing. If you get a spade, you win $$\$ 5$$. For any club, you win \(\$ 10\) plus an extra \(\$ 20\) for the ace of clubs. a. Create a probability model for the amount you win. b. Find the expected amount you'll win. c. What would you be willing to pay to play this game?

A delivery company's trucks occasionally get parking tickets, and based on past experience, the company plans that each truck will average 1.3 tickets a month, with a standard deviation of 0.7 tickets. a. If they have 18 trucks, what are the mean and standard deviation of the total number of parking tickets the company will have to pay this month? b. What assumption did you make in answering?

1 is 0.8 . If you get contract #1, the probability you also get contract #2 will be 0.鈥� # Your company bids for two contracts. You believe the probability you get contract #1 is 0.8 . If you get contract #1, the probability you also get contract #2 will be 0.2, and if you do not get #1, the probability you get #2 will be 0.3 a. Are the two contracts independent? Explain. b. Find the probability you get both contracts. c. Find the probability you get no contract. d. Let \(X\) be the number of contracts you get. Find the probability model for \(X\) e. Find the expected value and standard deviation.

Organizers of a televised fundraiser know from past experience that most people donate small amounts \((\$ 10-\$ 25),\) some donate larger amounts \((\$ 50-\$ 100),\) and \(a\) few people make very generous donations of \(\$ 250, \$ 500,\) or more. Historically, pledges average about \(\$ 32\) with a standard deviation of \(\$ 54\). a. If 120 people call in pledges, what are the mean and standard deviation of the total amount raised? b. What assumption did you make in answering this question?
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