Ch. 3 - Describing, Exploring, and Comparing Data

Triola - Elementary Statistics 14th Edition

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

Triola 14th Edition

Ch. 3 - Describing, Exploring, and Comparing Data

Problem 3.3.10

Chapter 3, Problem 3.3.10

Significant Values. In Exercises 9–12, use the range rule of thumb to identify (a) the values that are significantly low, (b) the values that are significantly high, and (c) the values that are neither significantly low nor significantly high.

IQ Scores The Wechsler test is used to measure intelligence of adults aged 16 to 80. The mean test score is 100 and the standard deviation is 15.

Verified step by step guidance

Step 1: Recall the range rule of thumb, which states that values are considered significantly low if they are below μ - 2σ and significantly high if they are above μ + 2σ, where μ is the mean and σ is the standard deviation.

Step 2: Substitute the given values into the formulas. The mean (μ) is 100, and the standard deviation (σ) is 15. Calculate the lower threshold for significantly low values using the formula μ - 2σ.

Step 3: Calculate the upper threshold for significantly high values using the formula μ + 2σ.

Step 4: Identify the range of values that are neither significantly low nor significantly high. These values fall between the lower threshold (μ - 2σ) and the upper threshold (μ + 2σ).

Step 5: Conclude by stating the thresholds for significantly low, significantly high, and the range of values that are neither, based on the calculations from the previous steps.

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Key Concepts

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

Mean

The mean is the average of a set of values, calculated by summing all the values and dividing by the number of values. In the context of IQ scores, the mean score of 100 indicates the central point around which the scores are distributed. Understanding the mean is crucial for identifying how individual scores compare to the average.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates 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. In this case, a standard deviation of 15 helps determine the range of scores that are considered typical or atypical.

Range Rule of Thumb

The range rule of thumb is a simple method for identifying significant values in a dataset. It suggests that values are significantly low if they fall below the mean minus two standard deviations and significantly high if they exceed the mean plus two standard deviations. This rule helps in categorizing IQ scores into significantly low, significantly high, and neither, based on the provided mean and standard deviation.

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Related Practice

Textbook Question

Finding Standard Deviation from a Frequency Distribution. In Exercises 37–40, refer to the frequency distribution in the given exercise and compute the standard deviation by using the formula below, where x represents the class midpoint, f represents the class frequency, and n represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 3-4 with the original list of data values: (Exercise 37) 18.5 minutes; (Exercise 38) 36.7 minutes; (Exercise 39) 6.9 years; (Exercise 40) 20.4 seconds.

Standard deviation for frequency distribution

Textbook Question

Comparing Values. In Exercises 13–16, use z scores to compare the given values.

Tallest and Shortest Men The tallest adult male was Robert Wadlow, and his height was 272 cm. The shortest adult male was Chandra Bahadur Dangi, and his height was 54.6 cm. Heights of men have a mean of 174.12 cm and a standard deviation of 7.10 cm. Which of these two men has the height that is more extreme?

Textbook Question

In Exercises 21–28, use the same list of cell phone radiation levels given for Exercises 17–20. Find the indicated percentile or quartile.

Textbook Question

z Scores If your score on your next statistics test is converted to a z score, which of these z scores would you prefer: -2.00, -1.00, 0, 1.00, 2.00? Why?

Textbook Question

In Exercises 29–32, compute the mean of the data summarized in the frequency distribution. Also, compare the computed means to the actual means obtained by using the original list of data values, which are as follows: (29) 31.4 minutes; (Exercise 30) 140.6 minutes; (Exercise 31) 55.2 years; (Exercise 32) 240.2 seconds.

Textbook Question

Estimating Standard Deviation with the Range Rule of Thumb. In Exercises 29–32, refer to the data in the indicated exercise. After finding the range of the data, use the range rule of thumb to estimate the value of the standard deviation. Compare the result to the standard deviation computed using all of the data.

Audiometry Use the hearing measurements from Data Set 7 “Audiometry.” Does it appear that the amounts of variation are different for the right threshold measurements and the left threshold measurements?