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 6.3.1a

Chapter 6, Problem 6.3.1a

Fatal Car Crashes There are about 15,000 car crashes each day in the United States, and the proportion of car crashes that are fatal is 0.00559 (based on data from the National Highway Traffic Safety Administration). Assume that each day, 1000 car crashes are randomly selected and the proportion of fatal car crashes is recorded.
a. What do you know about the mean of the sample proportions?

Verified step by step guidance

The mean of the sample proportions is a measure of the central tendency of the sample proportions. According to the Central Limit Theorem, the mean of the sample proportions is equal to the population proportion (p).

In this case, the population proportion (p) is given as 0.00559, which represents the proportion of car crashes that are fatal.

Thus, the mean of the sample proportions (denoted as μ_p̂) is equal to the population proportion: μ_p̂ = p.

This result holds because the sampling distribution of the sample proportion is centered around the true population proportion when the samples are randomly selected.

To summarize, the mean of the sample proportions is μ_p̂ = 0.00559, which is the same as the population proportion.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Sample Proportion

The sample proportion is the ratio of the number of successes (in this case, fatal car crashes) to the total number of observations in a sample. It is denoted as p̂ and provides an estimate of the population proportion. Understanding sample proportions is crucial for making inferences about the population based on sample data.

Mean of Sample Proportions

The mean of sample proportions refers to the expected value of the sample proportion across many samples. According to the Central Limit Theorem, this mean will be equal to the true population proportion (0.00559 in this case) when the sample size is sufficiently large, allowing for reliable estimates of the population parameter.

Central Limit Theorem

The Central Limit Theorem states that the distribution of the sample means (or sample proportions) will approach a normal distribution as the sample size increases, regardless of the original distribution of the population. This theorem is fundamental in statistics as it justifies the use of normal probability models for inference when dealing with large samples.

Recommended video:

Guided course

04:52

Calculating the Mean

Related Practice

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.

a. Given that the water taxi that sank was rated for a load limit of 3500 lb, what is the maximum mean weight of the passengers if the boat is filled to the stated capacity of 25 passengers?

views

Textbook Question

Body Temperatures Listed below are body temperatures (°F) of adult males (based on Data Set 5 “Body Temperatures” in Appendix B).

97.6 98.2 99.6 98.7 99.4 98.2 98.0 98.6 98.6

a. Find the mean. Does the result seem reasonable?

130

views

Textbook Question

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

Designing Manholes According to the website www.torchmate.com, “manhole covers must be a minimum of 22 in. in diameter, but can be as much as 60 in. in diameter.” Assume that a manhole is constructed to have a circular opening with a diameter of 22 in. Men have shoulder widths that are normally distributed with a mean of 18.2 in. and a standard deviation of 1.0 in. (based on data from the National Health and Nutrition Examination Survey).

a. What percentage of men will fit into the manhole?

118

views

Textbook Question

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.

a. If she curves by adding 15 to each grade, what is the new mean and standard deviation?

125

views

Textbook Question

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.

a. What are the values of the mean and standard deviation after converting all weights of Hershey Kisses to z scores using z = (x - μ)/σ ?

b. The original weights are in grams. What are the units of the corresponding z scores?

211

views

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.

views