Ch. 6 - Normal Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 6 - Normal Probability Distributions

Problem 6.1.48

Chapter 6, Problem 6.1.48

Basis for the Range Rule of Thumb and the Empirical Rule. In Exercises 45–48, find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.

About __ % of the area is between z = -3.5 and z = 3.5 (or within 3.5 standard deviation of the mean).

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the area under the standard normal distribution curve between z = -3.5 and z = 3.5. This area represents the proportion of data within 3.5 standard deviations of the mean.

Step 2: Recall that the standard normal distribution is symmetric about the mean (z = 0). The total area under the curve is 1, which corresponds to 100%.

Step 3: Use a standard normal distribution table (z-table) or statistical software to find the cumulative area to the left of z = 3.5. Similarly, find the cumulative area to the left of z = -3.5.

Step 4: Subtract the cumulative area to the left of z = -3.5 from the cumulative area to the left of z = 3.5. This difference gives the area between z = -3.5 and z = 3.5.

Step 5: Convert the area obtained in Step 4 into a percentage by multiplying it by 100. This percentage represents the proportion of the data within 3.5 standard deviations of the mean.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Normal Distribution

The standard normal distribution is a special normal distribution with a mean of 0 and a standard deviation of 1. It is used to determine probabilities and areas under the curve for any normal distribution by converting raw scores (z-scores) into standard scores. This allows for the comparison of different datasets and the application of statistical rules, such as the Empirical Rule.

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule helps in understanding the spread of data and is foundational for making predictions about probabilities in a normal distribution.

Range Rule of Thumb

The Range Rule of Thumb is a guideline that suggests the range of a dataset can be estimated as four times the standard deviation. This rule provides a quick way to assess the variability of data and is particularly useful in applied statistics for making rough estimates about the spread of data points in relation to the mean.

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Textbook Question

Standard Normal Distribution. In Exercises 13–16, find the indicated z score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

Textbook Question

Outliers For the purposes of constructing modified boxplots as described in Section 3-3, outliers are defined as data values that are above Q3 by an amount greater than 1.5 x IQR or below Q1 by an amount greater than 1.5 x IQR, where IQR is the interquartile range. Using this definition of outliers, find the probability that when a value is randomly selected from a normal distribution, it is an outlier.

Textbook Question

Determining Normality. In Exercises 9–12, refer to the indicated sample data and determine whether they appear to be from a population with a normal distribution. Assume that this requirement is loose in the sense that the population distribution need not be exactly normal, but it must be a distribution that is roughly bell-shaped.

Taxi Trips The distances (miles) traveled by New York City taxis transporting customers, as listed in Data Set 32 “Taxis” in Appendix B

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Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).

35 35 20 50 95 75 45 50 30 35 30 30

h. Are the wait times discrete data or continuous data?

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Textbook Question

Foot Lengths of Women Assume that foot lengths of adult females are normally distributed with a mean of 246.3 mm and a standard deviation of 12.4 mm (based on Data Set 3 “ANSUR II 2012” in Appendix B).

a. Find the probability that a randomly selected adult female has a foot length less than 221.5 mm.

Textbook Question

c. Find P95.