Skip to main content
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 6.6.20a
Chapter 6, Problem 6.6.20a

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?

Verified step by step guidance
1
Step 1: Identify the type of probability distribution to use. Since we are dealing with a large sample size (420,095 people) and a rare event (developing cancer), we can approximate the binomial distribution with a normal distribution. The binomial distribution has parameters n (number of trials) and p (probability of success). Here, n = 420,095 and p = 0.000340.
Step 2: Calculate the mean (μ) and standard deviation (σ) of the binomial distribution. The mean is given by μ = n * p, and the standard deviation is given by σ = sqrt(n * p * (1 - p)). Use these formulas to compute the mean and standard deviation.
Step 3: Convert the discrete binomial problem to a continuous normal approximation. To do this, apply the continuity correction by adjusting the value of 135 to 135.5 (adding 0.5 to account for the discrete-to-continuous transition).
Step 4: Standardize the value of 135.5 to a z-score using the formula z = (x - μ) / σ, where x is the value of interest (135.5), μ is the mean, and σ is the standard deviation. This z-score will allow us to find the cumulative probability using the standard normal distribution.
Step 5: Use a standard normal distribution table or a statistical software to find the cumulative probability corresponding to the calculated z-score. This cumulative probability represents the probability of 135 or fewer cases of cancer in the group. Compare this probability to the expected value (143 cases) to interpret whether the observed value (135 cases) is significantly different, and discuss whether the results support or refute the media reports about cell phones causing cancer.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Probability Distribution

A probability distribution describes how the probabilities of a random variable are distributed across its possible values. In this context, we can use the binomial distribution to model the number of cancer cases among cell phone users, where each user has a fixed probability of developing cancer. Understanding this distribution is crucial for calculating the likelihood of observing 135 or fewer cases in the sample of 420,095.
Recommended video:
Guided course
06:39
Calculating Probabilities in a Binomial Distribution

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. In this scenario, we can formulate a null hypothesis that cell phone usage does not affect the probability of developing brain cancer. By comparing the observed number of cases (135) to the expected number (143), we can assess whether the difference is statistically significant, which informs the discussion on media reports regarding cell phone safety.
Recommended video:
Guided course
06:21
Step 1: Write Hypotheses

Statistical Significance

Statistical significance indicates whether the results observed in a study are likely due to chance or if they reflect a true effect. In this case, determining if the number of brain cancer cases among cell phone users is significantly different from what would be expected based on the general population's cancer rate is essential. If the observed cases are not statistically significant, it suggests that the media reports may overstate the risk associated with cell phone usage.
Recommended video:
Guided course
05:53
Parameters vs. Statistics
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.

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 gains at least 2.0 kg during his freshman year..)

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.


b. Find the probability that a sample of 27 randomly selected adult males has a mean weight greater than 148 lb.

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

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


a. For the population, find the proportion of odd numbers.

Textbook Question

Smartphones Based on an LG smartphone survey, assume that 51% of adults with smartphones use them in theaters. In a separate survey of 250 adults with smartphones, it is found that 109 use them in theaters.


a. If the 51% rate is correct, find the probability of getting 109 or fewer smartphone owners who use them in theaters.

1
views