Ch. 6 - Normal Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446당신이 사용하는 게 아니라요?교과서 변경

Triola 14th Edition

Ch. 6 - Normal Probability Distributions

문제 6.1.25

6장, 문제 6.1.25

Standard Normal Distribution. In Exercises 17–36, assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.

Between 1.50 and 2.00

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Step 1: Understand the problem. The bone density test scores follow a standard normal distribution, which means the mean (μ) is 0 and the standard deviation (σ) is 1. We are tasked with finding the probability that a score lies between 1.50 and 2.00.

Step 2: Represent the problem graphically. Draw a standard normal distribution curve (bell-shaped curve) with the mean at 0. Mark the points 1.50 and 2.00 on the horizontal axis. Shade the area under the curve between these two points, as this represents the probability we are trying to find.

Step 3: Use the cumulative distribution function (CDF) of the standard normal distribution to find the probabilities corresponding to the z-scores of 1.50 and 2.00. The CDF gives the probability that a value is less than or equal to a given z-score. Denote these probabilities as P(Z ≤ 2.00) and P(Z ≤ 1.50).

Step 4: Calculate the probability of the range by subtracting the smaller cumulative probability from the larger one. Specifically, compute P(1.50 ≤ Z ≤ 2.00) = P(Z ≤ 2.00) - P(Z ≤ 1.50).

Step 5: If using technology (e.g., a calculator or statistical software), input the z-scores 1.50 and 2.00 into the standard normal distribution function to find the corresponding cumulative probabilities. Subtract the results as described in Step 4 to obtain the final probability.

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주요 개념

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Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It is represented by the Z-score, which indicates how many standard deviations an element is from the mean. This distribution is symmetric and bell-shaped, making it useful for calculating probabilities and percentiles for normally distributed data.

Z-scores

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z-scores allow for the comparison of scores from different normal distributions by standardizing them, making it easier to find probabilities using the standard normal distribution.

Probability and Area Under the Curve

In the context of the normal distribution, the probability of a score falling within a certain range is represented by the area under the curve of the distribution graph. To find this probability, one can use Z-scores to determine the corresponding areas from the standard normal distribution table or technology. The total area under the curve equals 1, representing the total probability of all possible outcomes.

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Car Colors

In Exercises 9–12, assume that 100 cars are randomly selected. Refer to the accompanying graph, which shows the top car colors and the percentages of cars with those colors (based on PPG Industries).

Black Cars Find the probability that at least 25 cars are black. Is 25 a significantly high number of black cars?

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Continuous Uniform Distribution. In Exercises 5–8, refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.

Less than 4.00 minutes

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Births: Sampling Distribution of Sample Proportion When two births are randomly selected, the sample space for genders is bb, bg, gb, and gg (where and Assume that those four outcomes are equally likely. Construct a table that describes the sampling distribution of the sample proportion of girls from two births. Does the mean of the sample proportions equal the proportion of girls in two births? Does the result suggest that a sample proportion is an unbiased estimator of a population proportion?

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Distributions In a continuous uniform distribution,

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a. Find the mean and standard deviation for the distribution of the waiting times represented in Figure 6-2, which accompanies Exercises 5–8.

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Constructing Normal Quantile Plots. In Exercises 17–20, use the given data values to identify the corresponding z scores that are used for a normal quantile plot, then identify the coordinates of each point in the normal quantile plot. Construct the normal quantile plot, then determine whether the data appear to be from a population with a normal distribution.

Earthquake Depths A sample of depths (km) of earthquakes is obtained from Data Set 24 “Earthquakes” in Appendix B: 17.3, 7.0, 7.0, 7.0, 8.1, 6.8.

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Finding Bone Density Scores. In Exercises 37–40 assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.

Find P99, the 99th percentile. This is the bone density score separating the bottom 99% from the top 1%.

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