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 6.2.5

Chapter 6, Problem 6.2.5

IQ Scores. In Exercises 5–8, find the area of the shaded region. The graphs depict IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler IQ test).

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Step 1: Understand the problem. The graph represents a normal distribution of IQ scores with a mean (μ) of 100 and a standard deviation (σ) of 15. The shaded region corresponds to scores less than 118. We need to find the area of this shaded region, which represents the cumulative probability up to 118.

Step 2: Convert the IQ score of 118 into a z-score using the formula: z = (X - μ) / σ, where X is the IQ score, μ is the mean, and σ is the standard deviation. Substitute the values: X = 118, μ = 100, and σ = 15.

Step 3: Once the z-score is calculated, use a standard normal distribution table (z-table) or a statistical software/tool to find the cumulative probability corresponding to this z-score. This cumulative probability represents the area under the curve to the left of the z-score.

Step 4: Interpret the cumulative probability. The value obtained from the z-table or software is the proportion of the population with IQ scores less than 118.

Step 5: If required, express the cumulative probability as a percentage by multiplying the value by 100. This percentage represents the likelihood of an individual having an IQ score less than 118.

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Key Concepts

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

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.

Z-Scores

A Z-score indicates how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the score and then dividing by the standard deviation. For the IQ score of 118, the Z-score can be calculated to determine its position relative to the mean, which is essential for finding the area under the curve.

Area Under the Curve

The area under the curve in a normal distribution represents the probability of a score falling within a certain range. To find the area to the left of a specific score, such as 118, one can use Z-scores and standard normal distribution tables or software. This area corresponds to the proportion of the population that scores below that value.

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Textbook Question

Small Sample Weights of M&M plain candies are normally distributed. Twelve M&M plain candies are randomly selected and weighed, and then the mean of this sample is calculated. Is it correct to conclude that the resulting sample mean cannot be considered to be a value from a normally distributed population because the sample size of 12 is too small? Explain.

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Textbook Question

Tennis Replay In a recent year, there were 879 challenges made to referee calls in professional tennis singles play. Among those challenges, 231 challenges were upheld with the call overturned. Assume that in general, 25% of the challenges are successfully upheld with the call overturned.

a. If the 25% rate is correct, find the probability that among the 879 challenges, the number of overturned calls is exactly 231.

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Textbook Question

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 = -1 and z = 1 (or within 1 standard deviation of the mean).

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