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Chapter 7: Problem 10

The mean of a normal probability distribution is \(60 ;\) the standard deviation is \(5 .\) a. About what percent of the observations lie between 55 and \(65 ?\) b. About what percent of the observations lie between 50 and \(70 ?\) c. About what percent of the observations lie between 45 and \(75 ?\)

Short Answer

Expert verified
a) 68% b) 95% c) 99.7%

Step by step solution

01

Understanding the Normal Distribution

We have a normal distribution with a mean \( \mu = 60 \) and a standard deviation \( \sigma = 5 \). We are to find the percentage of observations between certain values using the properties of a normal distribution.
02

Use Empirical Rule for 55 to 65

The empirical rule states that about 68% of the data falls within one standard deviation of the mean in a normal distribution. Here, one standard deviation from the mean \( 60 \) is \([60 - 5, 60 + 5] = [55, 65]\). Thus, about 68% of the observations lie between 55 and 65.
03

Use Empirical Rule for 50 to 70

Two standard deviations from the mean are \( \mu \pm 2\sigma = 60 \pm (2 \times 5) = [50, 70] \). The empirical rule suggests that about 95% of observations lie within two standard deviations. Thus, about 95% of the observations lie between 50 and 70.
04

Use Empirical Rule for 45 to 75

Three standard deviations from the mean are \( \mu \pm 3\sigma = 60 \pm (3 \times 5) = [45, 75] \). The empirical rule indicates that about 99.7% of observations fall within three standard deviations. Therefore, about 99.7% of the observations lie between 45 and 75.

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Key Concepts

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

Understanding the Empirical Rule
The empirical rule is a simple guideline to understand how data in a normal distribution are spread out around the mean. It's like a handy tool for quickly estimating the spread of values. In a normal distribution:
  • About 68% of the data falls within one standard deviation (蟽) from the mean (渭).
  • Approximately 95% lies within two standard deviations.
  • Nearly 99.7% is enclosed within three standard deviations.
Knowing this rule helps predict the percentage of observations within specific intervals in a bell-shaped, normally distributed dataset.
For instance, if you're examining test scores or measuring product weights that follow a normal distribution, the empirical rule allows you to quickly find out how concentrated the observations are around the mean. This makes it a valuable tool for data analysis and statistics.
The Mean and Standard Deviation
The mean and standard deviation are central to understanding any normal distribution. The mean, often symbolized as 渭, represents the average value of all the data points. It's the "center" of the normal distribution.
Imagine calculating the average score from a set of test results; that is what the mean tells you. On the other hand, the standard deviation, denoted by 蟽, measures the spread or dispersion of data around the mean.
Here's a simple way to think about it:
  • If the standard deviation is small, the data are closely clustered around the mean, which implies less variability.
  • A large standard deviation indicates that the data points are spread over a wider range, suggesting more variability.
In practical terms, having a good grasp of mean and standard deviation helps in predicting outcomes and understanding the distribution's shape and spread.
Percentage of Observations in Intervals
Using the empirical rule and knowing the mean and standard deviation, you can determine what percentage of observations fall within specific intervals. For example, if the mean is 60 and the standard deviation is 5:
  • Between 55 and 65: This range represents one standard deviation from the mean. Based on the empirical rule, roughly 68% of observations fall within this range.
  • Between 50 and 70: This interval covers two standard deviations. Here, about 95% of observations are included, as outlined by the empirical rule.
  • Between 45 and 75: Spanning three standard deviations, this interval contains about 99.7% of observations.
Understanding these percentages helps in making informed decisions in fields like quality control, where predicting product consistency within certain bounds is crucial. It's about seeing where most of the data falls within a normal distribution.

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Most popular questions from this chapter

A normal population has a mean of 20.0 and a standard deviation of \(4.0 .\) a. Compute the \(z\) value associated with 25.0 . b. What proportion of the population is between 20.0 and \(25.0 ?\) c. What proportion of the population is less than \(18.0 ?\)

In establishing warranties on HDTV sets, the manufacturer wants to set the limits so that few will need repair at manufacturer expense. On the other hand, the warranty period must be long enough to make the purchase attractive to the buyer. For a new HDTV the mean number of months until repairs are needed is 36.84 with a standard deviation of 3.34 months. Where should the warranty limits be set so that only 10 percent of the HDTVs need repairs at the manufacturer's expense?

A uniform distribution is defined over the interval from 6 to \(10 .\) a. What are the values for a and \(b\) ? b. What is the mean of this uniform distribution? c. What is the standard deviation? d. Show that the total area is 1.00 . e. Find the probability of a value more than 7 . f. Find the probability of a value between 7 and \(9 .\)

The current model Boeing 737 has a capacity of 189 passengers. Suppose Delta Airlines uses this equipment for its Atlanta to Houston flights. The distribution of the number of seats sold for the Atlanta to Houston flights follows the normal distribution with a mean of 155 seats and a standard deviation of 15 seats. a. What is the likelihood of selling more than 134 seats? b. What is the likelihood of selling less than 173 seats? c. What is the likelihood of selling more than 134 seats but less than 173 seats? d. What percent of the time would Delta be able to sell more seats than there are seats actually available?

The weights of canned hams processed at the Henline Ham Company follow the normal distribution, with a mean of 9.20 pounds and a standard deviation of 0.25 pounds. The label weight is given as 9.00 pounds. a. What proportion of the hams actually weigh less than the amount claimed on the label? b. The owner, Glen Henline, is considering two proposals to reduce the proportion of hams below label weight. He can increase the mean weight to 9.25 and leave the standard deviation the same, or he can leave the mean weight at 9.20 and reduce the standard deviation from 0.25 pounds to \(0.15 .\) Which change would you recommend?
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