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Ch. 6 - Normal Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 6 - Normal Probability DistributionsProblem 6.2.37c
Chapter 6, Problem 6.2.37c

Curving Test Scores A professor gives a test and the scores are normally distributed with a mean of 60 and a standard deviation of 12. She plans to curve the scores.


c. If the grades are curved so that grades of B are given to scores above the bottom 70% and below the top 10%, find the numerical limits for a grade of B.

Verified step by step guidance
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Step 1: Understand the problem. The scores are normally distributed with a mean (μ) of 60 and a standard deviation (σ) of 12. We need to find the numerical limits for a grade of B, which corresponds to scores above the bottom 70% and below the top 10%. This means we are looking for the scores corresponding to the 70th percentile and the 90th percentile of the normal distribution.
Step 2: Convert the given percentiles (70% and 90%) into z-scores using the standard normal distribution table or a z-score calculator. Recall that the z-score represents the number of standard deviations a value is from the mean. For example, the z-score for the 70th percentile is the value of z such that the cumulative probability up to z is 0.70.
Step 3: Use the z-score formula to convert the z-scores into raw scores (X) in the context of the given normal distribution. The formula is: X=μ+zσ, where μ is the mean, σ is the standard deviation, and z is the z-score.
Step 4: Substitute the mean (μ = 60), standard deviation (σ = 12), and the z-scores for the 70th and 90th percentiles into the formula to calculate the corresponding raw scores. These raw scores will represent the numerical limits for a grade of B.
Step 5: Interpret the results. The lower limit for a grade of B is the raw score corresponding to the 70th percentile, and the upper limit is the raw score corresponding to the 90th percentile. These values define the range of scores that will receive a grade of B.

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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, showing that data near the mean are more frequent in occurrence than data far from the mean. In this context, the test scores follow a normal distribution with a specified mean and standard deviation, which allows us to use statistical methods to determine the limits for grades.
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Finding Standard Normal Probabilities using z-Table

Percentiles

Percentiles are measures that indicate the value below which a given percentage of observations in a group of observations falls. For this question, we need to find the scores that correspond to the 30th percentile (bottom 70%) and the 90th percentile (top 10%) of the distribution to determine the numerical limits for a grade of B.

Z-scores

A Z-score represents the number of standard deviations a data point is from the mean. It is calculated by subtracting the mean from the score and dividing by the standard deviation. In this scenario, Z-scores will help us convert the percentiles into actual test scores, allowing us to find the specific score limits for a grade of B.
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Z-Scores From Given Probability - TI-84 (CE) Calculator
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


c. Find the standard deviation s.

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


g. What level of measurement (nominal, ordinal, interval, ratio) describes this data set?

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Textbook Question

In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement.


c. Find the mean of the sampling distribution of the sample variance.

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Textbook Question

Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem.


Safe Loading of Elevators The elevator in the car rental building at San Francisco International Airport has a placard stating that the maximum capacity is “4000 lb—27 passengers.” Because 4000/27=148, this converts to a mean passenger weight of 148 lb when the elevator is full. We will assume a worst-case scenario in which the elevator is filled with 27 adult males. Based on Data Set 1 “Body Data” in Appendix B, assume that adult males have weights that are normally distributed with a mean of 189 lb and a standard deviation of 39 lb.


c. What do you conclude about the safety of this elevator?

Textbook Question

In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement.


Sampling Distribution of the Sample Proportion


c. Find the mean of the sampling distribution of the sample proportion of odd numbers.

Textbook Question

Hybridization A hybridization experiment begins with four peas having yellow pods and one pea having a green pod. Two of the peas are randomly selected with replacement from this population.


c. Is the mean of the sampling distribution [from part (b)] equal to the population proportion of peas with yellow pods? Does the mean of the sampling distribution of proportions always equal the population proportion?