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.CRE.1e

Chapter 6, Problem 6.CRE.1e

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

e. Convert the longest wait time to a z score.
f. Based on the result from part (e), is the longest wait time significantly high?

Verified step by step guidance

Step 1: Identify the longest wait time in the data set. The data set is: 35, 35, 20, 50, 95, 75, 45, 50, 30, 35, 30, 30. The longest wait time is 95 minutes.

Step 2: Calculate the mean (μ) of the data set. To do this, sum all the wait times and divide by the total number of data points. Use the formula:

μ = \frac{\sum x}{n}

, where ∑x is the sum of the data points and n is the number of data points.

Step 3: Calculate the standard deviation (σ) of the data set. Use the formula:

σ = \sqrt{\frac{\sum (^{x − μ) 2}}{n}}

, where x represents each data point, μ is the mean, and n is the number of data points.

Step 4: Convert the longest wait time (95 minutes) to a z-score using the formula:

z = \frac{x − μ}{σ}

, where x is the data point (95 minutes), μ is the mean, and σ is the standard deviation.

Step 5: Determine if the z-score indicates that the longest wait time is significantly high. A z-score is typically considered significantly high if it is greater than 2. Interpret the z-score value to answer part (f).

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Key Concepts

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

Z-Score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It indicates how many standard deviations an element is from the mean. A positive Z-score means the value is above the mean, while a negative Z-score indicates it is below. This concept is crucial for determining how unusual or typical a particular data point is within a dataset.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. Understanding standard deviation is essential for calculating Z-scores and assessing the significance of data points in relation to the overall dataset.

Significance in Statistics

In statistics, significance refers to the likelihood that a result or relationship is caused by something other than mere random chance. A common threshold for significance is a p-value of less than 0.05. In the context of the longest wait time, determining if it is significantly high involves comparing its Z-score to critical values that indicate whether it falls within a typical range or is an outlier.

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Related Practice

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

d. The accompanying normal quantile plot is obtained by using all 50 wait times at 10:00 AM for the Tower of Terror ride at Disney World. Based on this normal quantile plot, do the sample data appear to be from a normally distributed population?

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Textbook Question

Mensa Membership in Mensa requires a score in the top 2% on a standard intelligence test. The Wechsler IQ test is designed for a mean of 100 and a standard deviation of 15, and scores are normally distributed.

b. If 4 randomly selected adults take the Wechsler IQ test, find the probability that their mean score is at least 131.

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Textbook Question

Blue Eyes Assume that 35% of us have blue eyes (based on a study by Dr. P. Soria at Indiana University).

c. Find the probability of randomly selecting three different people and finding that all of them have blue eyes.

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Textbook Question

Blue Eyes Assume that 35% of us have blue eyes (based on a study by Dr. P. Soria at Indiana University).

b. Find the value of P(B_bar).

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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

a. Find the mean xbar.

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Textbook Question

a. Find the minimum Wechsler IQ test score that satisfies the Mensa requirement.

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