Ch. 6 - Normal Probability Distributions

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 6 - Normal Probability Distributions

Problem 6.4.6a

Chapter 6, Problem 6.4.6a

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).

a. If 1 male college student is randomly selected, find the probability that he gains at least 2.0 kg during his freshman year..)

Verified step by step guidance

Step 1: Identify the given parameters. From the problem, the mean (μ) is 1.2 kg, the standard deviation (σ) is 4.9 kg, and we are looking for the probability that a randomly selected male college student gains at least 2.0 kg. This means we need to calculate P(X ≥ 2.0), where X is the weight gain.

Step 2: Standardize the value of 2.0 kg using the z-score formula. The z-score formula is given by:

z = (X - μ) / σ

. Substitute the values X = 2.0, μ = 1.2, and σ = 4.9 into the formula.

Step 3: Simplify the z-score calculation to find the standardized z-value. This will give you the z-score corresponding to the weight gain of 2.0 kg.

Step 4: Use the standard normal distribution table (or a statistical software/tool) to find the cumulative probability corresponding to the calculated z-score. This cumulative probability represents P(X ≤ 2.0).

Step 5: Since the problem asks for P(X ≥ 2.0), use the complement rule: P(X ≥ 2.0) = 1 - P(X ≤ 2.0). Subtract the cumulative probability from 1 to find the desired probability.

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Key Concepts

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

Central Limit Theorem

The Central Limit Theorem (CLT) states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the original distribution of the population. This theorem is crucial for making inferences about population parameters based on sample statistics, especially when dealing with large samples.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the weights gained by male college students are normally distributed, which allows us to use properties of the normal distribution to calculate probabilities related to weight gain.

Z-Score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. It is calculated by subtracting the mean from the value and dividing by the standard deviation. Z-scores are essential for determining probabilities in a normal distribution, allowing us to find the likelihood of a specific weight gain.

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Related Practice

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 Standard Deviation For the following, round results to three decimal places.

a. Find the value of the population standard deviation σ.

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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.

a. After identifying the 25 different possible samples, find the proportion of peas with yellow pods in each of them, then construct a table to des

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Textbook Question

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

Redesign of Ejection Seats When women were finally allowed to become pilots of fighter jets, engineers needed to redesign the ejection seats because they had been originally designed for men only. The ACES-II ejection seats were designed for men weighing between 140 lb and 211 lb. Weights of women are now normally distributed with a mean of 171 lb and a standard deviation of 46 lb (based on Data Set 1 “Body Data” in Appendix B).

a. If 1 woman is randomly selected, find the probability that her weight is between 140 lb and 211 lb.

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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.

Sampling Distribution of the Sample Median

a. Find the value of the population median.

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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).

a. If 1 male college student is randomly selected, find the probability that he has no weight gain during his freshman year. (That is, find the probability that during his freshman year, his weight gain is less than or equal to 0 kg.)

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Textbook Question

Cell Phones and Brain Cancer In a study of 420,095 cell phone users in Denmark, it was found that 135 developed cancer of the brain or nervous system. For those not using cell phones, there is a 0.000340 probability of a person developing cancer of the brain or nervous system. We therefore expect about 143 cases of such cancers in a group of 420,095 randomly selected people.

a. Find the probability of 135 or fewer cases of such cancers in a group of 420,095 people.

b. What do these results suggest about media reports that suggest cell phones cause cancer of the brain or nervous system?

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