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Chapter 7: Problem 9

Find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found. Determine the area under the standard normal curve that lies between (a) \(Z=-2.04\) and \(Z=2.04\) (b) \(Z=-0.55\) and \(Z=0\) (c) \(Z=-1.04\) and \(Z=2.76\)

Short Answer

Expert verified
(a) 0.9586, (b) 0.2088, (c) 0.8479

Step by step solution

01

Understanding the Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Areas under this curve represent probabilities.
02

Identify the Areas for Each Z-Score Range

Determine the range of Z-scores for each part of the problem: (a) from -2.04 to 2.04, (b) from -0.55 to 0, and (c) from -1.04 to 2.76.
03

Step (a): Area Between Z = -2.04 and Z = 2.04

Use the standard normal distribution table or a calculator to find the cumulative areas: Area to the left of Z = 2.04: 0.9793 Area to the left of Z = -2.04: 0.0207 Subtract the smaller area from the larger area: 0.9793 - 0.0207 = 0.9586
04

Step (b): Area Between Z = -0.55 and Z = 0

Use the standard normal distribution table or a calculator: Area to the left of Z = 0: 0.5 (since 0 is the mean) Area to the left of Z = -0.55: 0.2912 Subtract the smaller area from the larger area: 0.5 - 0.2912 = 0.2088
05

Step (c): Area Between Z = -1.04 and Z = 2.76

Use the standard normal distribution table or a calculator: Area to the left of Z = 2.76: 0.9971 Area to the left of Z = -1.04: 0.1492 Subtract the smaller area from the larger area: 0.9971 - 0.1492 = 0.8479

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Z-scores
Understanding Z-scores is crucial when dealing with the standard normal distribution. A Z-score indicates how many standard deviations an element is from the mean.
Z-scores can be positive or negative:
  • Positive Z-score: The value is above the mean.
  • Negative Z-score: The value is below the mean.
A mean is 0, and a standard deviation is 1 in a standard normal distribution.
For example, a Z-score of 2.04 means the data point is 2.04 standard deviations above the mean. Similarly, a Z-score of -2.04 means it is 2.04 standard deviations below the mean.
cumulative areas
Cumulative areas represent the probability that a random variable will take a value less than or equal to a specific Z-score.
For example, the area to the left of Z=2.04 on the standard normal curve is 0.9793, meaning there's a 97.93% chance that a value is less than or equal to 2.04.
Knowing the cumulative area helps to determine the probability:
  • Use standard normal distribution tables or calculators to find cumulative areas.
  • Always remember that the total area under the standard normal curve is 1 (or 100%).
In practice, if finding the area between two Z-scores, you subtract the cumulative area of the lower Z-score from the cumulative area of the higher Z-score.
probability under the curve
Probability under the curve is the area between two Z-scores on the standard normal distribution.
This area represents the likelihood of a value falling within that range.
  • For instance, between Z=-2.04 and Z=2.04, the area is 0.9586. So, there's a 95.86% probability of a value being in this range.
  • Similarly, between Z=-0.55 and Z=0, the area is 0.2088, or 20.88% probability.
  • For Z=-1.04 to Z=2.76, the area is 0.8479, or 84.79% probability.
Understanding these probabilities is vital in statistics, especially when assessing the likelihood of events.
normal distribution
The normal distribution is a probability distribution that is symmetrical about the mean, depicting data where most values cluster around a central point and taper off as they go further away from the mean.
Key characteristics include:
  • Bell-shaped curve: Peaks at the mean (0 in a standard normal distribution).
  • Mean, median, and mode are all equal.
  • About 68% of data falls within one standard deviation (Z-scores between -1 and 1).
  • About 95% of data falls within two standard deviations (Z-scores between -2 and 2).
  • Nearly all data (99.7%) falls within three standard deviations (Z-scores between -3 and 3).
Understanding the normal distribution helps in numerous statistical tasks, such as hypothesis testing, establishing confidence intervals, and data analysis.

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Most popular questions from this chapter

According to a report by the Commerce Department in the fall of \(2004,20 \%\) of U.S. households had some type of high-speed Internet connection. Suppose 80 U.S. households are selected at random. Use the normal approximation to the binomial to (a) approximate the probability that exactly 15 households have high-speed Internet access. (b) approximate the probability that at least 20 households have high-speed Internet access. (c) approximate the probability that fewer than 10 households have high-speed Internet access. (d) approximate the probability that between 12 and 18 households, inclusive, have high-speed Internet access.

Find the indicated probability of the standard normal random variable \(Z\). $$P(Z \geq 1.84)$$

Times The mean of the incubation time of fertilized chicken eggs kept at \(100.5^{\circ} \mathrm{F}\) in a still-air incubator is 21 days. Suppose the incubation times are approximately normally distributed with a standard deviation of 1 day. (Source: University of Illinois Extension.) (a) Determine the 17 th percentile for incubation times of fertilized chicken eggs. (b) Determine the incubation times that make up the middle \(95 \%\) of fertilized chicken eggs?

A discrete random variable is given. Assume the probability of the random variable will be approximated using the normal distribution. Describe the area under the normal curve that will be computed. For example, if we wish to compute the probability of finding at least five defective items in a shipment, we would approximate the probability by computing the area under the normal curve to the right of \(X=4.5\). The probability that fewer than 35 people support the privatization of Social Security.

Find the indicated \(Z\) -score. Be sure to draw a standard normal curve that depicts the solution. Find the \(Z\) -score such that the area under the standard normal curve to the left is 0.1.
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