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Chapter 2: Problem 42

Potato chips The distribution of weights of 9 -ounce bags of a particular brand of potato chips is approximately Normal with mean \(\mu=9.12\) ounces and standard deviation \(\sigma=0.05\) ounce. Draw an accurate sketch of the distribution of potato chip bag weights. Be sure to label the mean, as well as the points \(1,2,\) and 3 standard deviations away from the mean on the horizontal axis.

Short Answer

Expert verified
Sketch a normal curve with 9.12 at the center and key points at 9.07, 9.17, 9.02, 9.22, 8.97, and 9.27.

Step by step solution

01

Understand the Problem

The exercise asks us to graph a normal distribution for the weights of potato chip bags. The given distribution has a mean \( \mu = 9.12 \) ounces and a standard deviation \( \sigma = 0.05 \) ounces. Our goal is to mark the mean and the points that are 1, 2, and 3 standard deviations away from the mean on the horizontal axis.
02

Calculate Key Points on the Horizontal Axis

To mark the key points on the graph, we calculate the mean and the points at 1, 2, and 3 standard deviations from the mean. The values are: - Mean: \( \mu = 9.12 \)- 1 Standard Deviation (\( \mu - \sigma \) and \( \mu + \sigma \)): \( 9.12 - 0.05 = 9.07 \); \( 9.12 + 0.05 = 9.17 \).- 2 Standard Deviations (\( \mu - 2\sigma \) and \( \mu + 2\sigma \)): \( 9.12 - 2 \times 0.05 = 9.02 \); \( 9.12 + 2 \times 0.05 = 9.22 \).- 3 Standard Deviations (\( \mu - 3\sigma \) and \( \mu + 3\sigma \)): \( 9.12 - 3 \times 0.05 = 8.97 \); \( 9.12 + 3 \times 0.05 = 9.27 \).
03

Sketch the Normal Distribution

Draw a bell-shaped curve on a horizontal axis to represent the normal distribution. Label the mean at 9.12. Mark off intervals of 0.05 ounces away from the mean for 1, 2, and 3 standard deviations. Specifically, mark the points at 9.07 and 9.17 for 1 standard deviation, 9.02 and 9.22 for 2 standard deviations, and 8.97 and 9.27 for 3 standard deviations.

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Key Concepts

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

Mean and Standard Deviation
In any normal distribution, the mean and standard deviation are crucial concepts. The **mean** is essentially the "average" of all your data points. In the problem of potato chip bags, the mean weight, denoted as \( \mu \), is given as 9.12 ounces. This value represents the center point of the distribution, meaning that most bags will weigh around this amount.

The **standard deviation** is a measure of how "spread out" the values are from the mean. A smaller standard deviation means the values are closer to the mean, while a larger one indicates they are more spread out. For our potato chip problem, the standard deviation is \( \sigma = 0.05 \) ounces. This tells us that most weights will be within 0.05 ounces of the mean, showing a pretty tight distribution.

Both mean and standard deviation are key when analyzing the bell curve shape of a normal distribution. They help us predict how much of the data falls within certain ranges from the mean.
Statistical Graphing
Graphing is an excellent way to visualize how data is distributed across different values. For a normal distribution, we use a **bell curve** to represent how data combines around a mean value.

The bell curve essentially shows the probability of different outcomes. For the potato chip example, drawing the bell curve means plotting the mean at \( 9.12 \) ounces in the center. From there, you mark out 1, 2, and 3 standard deviations from both the left and right sides of the mean. This helps depict the spread of your data.

When sketching, it's essential to label each critical point:
  • 1 standard deviation away at \( 9.07 \) and \( 9.17 \)
  • 2 standard deviations at \( 9.02 \) and \( 9.22 \)
  • 3 standard deviations at \( 8.97 \) and \( 9.27 \)
This visual representation allows anyone to quickly see most bags' weights and how likely certain weights are to occur.
Calculating Standard Deviations
Knowing how to calculate standard deviations is useful when analyzing data. Let's delve into the procedure, using the exercise as an example.

You start with the mean and add or subtract multiples of the standard deviation to find key points on the normal distribution plot. These are marked at distances from the mean:
  • 1 standard deviation: Subtract \( \sigma = 0.05 \) from the mean (\( 9.12 - 0.05 \)) to get \( 9.07 \), and add \(\sigma\) to get \( 9.17 \).
  • 2 standard deviations: You'll double the standard deviation: subtract \( 2 \times 0.05 \) from \( 9.12 \) resulting in \( 9.02 \), or add it to land at \( 9.22 \).
  • 3 standard deviations: Tripling the standard deviation, subtract \( 3 \times 0.05 \) to get \( 8.97 \) and add it to reach \( 9.27 \).
These calculations help us know where most data or observations will fall relative to the mean. In a typical normal distribution, about 68% of data is within 1 standard deviation, 95% within 2, and 99.7% within 3. These insights are powerful when predicting outcomes based on statistical data.

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Most popular questions from this chapter

Deciles The deciles of any distribution are the values at the 10 th, \(20 t h, \ldots, 90\) th percentiles. The first and last deciles are the l0th and the 90 th percentiles, respectively. (a) What are the first and last deciles of the standard Normal distribution? (b) The heights of young women are approximately Normal with mean 64.5 inches and standard deviation 2.5 inches. What are the first and last deciles of this distribution? Show your work.

IQ test scores Scores on the Wechsler Adult Intelligence Scale (a standard IQ test) for the 20 to 34 age group are approximately Normally distributed with \(\mu=110\) and \(\sigma=25\) (a) At what percentile is an IQ score of \(150 ?\) (b) What percent of people aged 20 to 34 have IQs between 125 and \(150 ?\) (c) MENSA is an elite organization that admits as members people who score in the top \(2 \%\) on \(\mathrm{IQ}\) tests. What score on the Wechsler Adult Intelligence Scale would an individual aged 20 to 34 have to earn to qualify for MENSA membership?

Eleanor scores 680 on the SAT Mathematics test. The distribution of SAT scores is symmetric and single-peaked, with mean 500 and standard deviation 100 . Gerald takes the American College Testing (ACT) Mathematics test and scores 27. ACT scores also follow a symmetric, single-peaked distribution-but with mean 18 and standard deviation \(6 .\) Find the standardized scores for both students. Assuming that both tests measure the same kind of ability, who has the higher score?

I think I can! An important measure of the performance of a locomotive is its "adhesion," which is the locomotive's pulling force as a multiple of its weight. The adhesion of one 4400 -horsepower diesel locomotive varies in actual use according to a Normal distribution with mean \(\mu=0.37\) and standard deviation \(\sigma=0.04\) (a) For a certain small train's daily route, the locomotive needs to have an adhesion of at least 0.30 for the train to arrive at its destination on time. On what proportion of days will this happen? Show your method. (b) An adhesion greater than 0.50 for the locomotive will result in a problem because the train will arrive too early at a switch point along the route. On what proportion of days will this happen? Show your method. (c) Compare your answers to parts (a) and (b). Does it make sense to try to make one of these values larger than the other? Why or why not?

Put a lid on it! At some fast-food restaurants, customers who want a lid for their drinks get them from a large stack left near straws, napkins, and condiments. The lids are made with a small amount of flexibility so they can be stretched across the mouth of the cup and then snugly secured. When lids are too small or too large, customers can get very frustrated, especially if they end up spilling their drinks. At one particular restaurant, large drink cups require lids with a "diameter" of between 3.95 and 4.05 inches. The restaurant's lid supplier claims that the diameter of their large lids follows a Normal distribution with mean 3.98 inches and standard deviation 0.02 inches. Assume that the supplier's claim is true. (a) What percent of large lids are too small to fit? Show your method. (b) What percent of large lids are too big to fit? Show your method. (c) Compare your answers to parts (a) and (b). Does it make sense for the lid manufacturer to try to make one of these values larger than the other? Why or why not?
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