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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 6.2.2b
Chapter 6, Problem 6.2.2b

Hershey Kisses Based on Data Set 38 “Candies” in Appendix B, weights of the chocolate in Hershey Kisses are normally distributed with a mean of 4.5338 g and a standard deviation of 0.1039 g


b. What is the value of the median?

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1
Step 1: Understand the relationship between the mean and the median in a normal distribution. In a normal distribution, the mean, median, and mode are all equal because the distribution is symmetric.
Step 2: Identify the given parameters of the normal distribution. The problem states that the mean weight of Hershey Kisses is 4.5338 g and the standard deviation is 0.1039 g.
Step 3: Recall that the median in a normal distribution is equal to the mean. This is a property of the normal distribution due to its symmetry.
Step 4: Conclude that the median weight of Hershey Kisses is the same as the mean weight, which is 4.5338 g.
Step 5: No further calculations are needed because the median is directly determined by the mean in a normal distribution.

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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, indicating that data near the mean are more frequent in occurrence than data far from the mean. In a normal distribution, the mean, median, and mode are all equal, which simplifies the calculation of the median when the distribution is known.
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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 a normal distribution, the mean serves as the central point around which the data is distributed, and it is crucial for understanding the overall distribution of the dataset.
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Median

The median is the middle value of a dataset when it is ordered from least to greatest. For a normal distribution, the median is equal to the mean, making it straightforward to determine in this case. It represents the point at which half of the data points fall below and half fall above.
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Related Practice
Textbook Question

Eye Color Based on a study by Dr. P. Sorita at Indiana University, assume that 12% of us have green eyes. In a study of 650 people, it is found that 86 of them have green eyes.


b. Is 86 people with green eyes significantly high?

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Textbook Question

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


Water Taxi Safety Passengers died when a water taxi sank in Baltimore’s Inner Harbor. Men are typically heavier than women and children, so when loading a water taxi, assume a worst-case scenario in which all passengers are men. Assume that weights of men are normally distributed with a mean of 189 lb and a standard deviation of 39 lb (based on Data Set 1 “Body Data” in Appendix B). The water taxi that sank had a stated capacity of 25 passengers, and the boat was rated for a load limit of 3500 lb.


b. If the water taxi is filled with 25 randomly selected men, what is the probability that their mean weight exceeds the value from part (a)?

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Textbook Question

MCAT The Medical College Admissions Test (MCAT) is used to help screen applicants to medical schools. Like many such tests, the MCAT uses multiple-choice questions with each question having five choices, one of which is correct. Assume that you must make random guesses for two such questions. Assume that both questions have correct answers of “a.”


b. Find the mean of the sampling distribution of the sample proportion.

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Textbook Question

Using the Central Limit Theorem. In Exercises 5–8, assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of 1.2 kg and a standard deviation of 4.9 kg (based on Data Set 13 “Freshman 15” in Appendix B).


b. If 9 male college students are randomly selected, find the probability that their mean weight gain during freshman year is between 0 kg and 3 kg.

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Textbook Question

College Presidents There are about 4200 college presidents in the United States, and they have annual incomes with a distribution that is skewed instead of being normal. Many different samples of 40 college presidents are randomly selected, and the mean annual income is computed for each sample.


b. What value do the sample means target? That is, what is the mean of all such sample means?

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Textbook Question

Transformations The heights (in inches) of women listed in Data Set 1 “Body Data” in Appendix B have a distribution that is approximately normal, so it appears that those heights are from a normally distributed population.


b. If each height is converted from inches to centimeters, are the heights in centimeters also normally distributed?

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