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Chapter 6: Problem 32

Find the area under a normal distribution curve with \(\mu=18.3\) and \(\sigma=3.4\) a. to the left of \(x=10.9\) b. to the right of \(x=14\) c. to the left of \(x=22.7\) d. to the right of \(x=29.2\)

Short Answer

Expert verified
a. 0.0146, b. 0.8962, c. 0.9015, d. 0.0007

Step by step solution

01

Transformation for Problem a

Firstly, transform \(x=10.9\) to a standard normal distribution: calculate \(z = (x - \mu) / \sigma\). It gives, \(z = (10.9 - 18.3)/3.4 = -2.18\)
02

Calculation of Probability for Problem a

Using z-table, find the probability of \(P(Z< -2.18)\). The area to the left of \(z= -2.18\) on a z-table is 0.0146
03

Transformation for Problem b

Transform \(x=14\) to a standard normal distribution: calculate \(z = (14 - 18.3)/3.4 = -1.26\)
04

Calculation of Probability for Problem b

Find \(P(Z> -1.26)= 1 - P(Z < -1.26)\). The area to the left of \(z= -1.26\) is 0.1038. So, the area to the right is \(1 - 0.1038 = 0.8962\)
05

Transformation for Problem c

Transform \(x=22.7\) to a standard normal distribution: calculate \(z = (22.7 - 18.3)/3.4 = 1.29\)
06

Calculation of Probability for Problem c

Find \(P(Z< 1.29)\). Using a z-table, the area to the left of \(z= 1.29\) is 0.9015
07

Transformation for Problem d

Transform \(x=29.2\) to a standard normal distribution: calculate \(z = (29.2 - 18.3)/3.4 = 3.20\)
08

Calculation of Probability for Problem d

Find \(P(Z> 3.20) = 1 - P(Z<3.20)\). The area to the left of \(z= 3.20\) is 0.9993. Thus, the area to the right is \(1 -0.9993 = 0.0007\)

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Key Concepts

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

Standard Normal Distribution
The standard normal distribution is a special normal distribution with a mean (\(\mu\)) of zero and a standard deviation (\(\sigma\)) of one. It is a standardized version of the normal distribution and allows for comparison across different datasets. To convert a normal distribution to a standard normal distribution, a z-score is calculated using the formula:
  • \[ z = \frac{x - \mu}{\sigma} \]
where \( x \) is a value from the dataset, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. By using z-scores, any normal distribution can be transformed into a standard normal distribution. Then, probabilities and areas under the curve can be easily determined using standard tools like the z-table.
Z-Table
The z-table, also known as the standard normal probability table, is a mathematical table that provides the probability of a statistic being below, above, or between standard scores on the standard normal distribution. When you look up a z-score on the z-table, it gives you the area (probability) to the left of that score.
  • If you need the area to the right, you simply subtract the table value from 1.
  • If you're calculating probabilities between two z-scores, you'd need the areas to the left of both, then subtract the smaller from the larger.
The z-table has two commonly used versions: the left-tail and right-tail tables. Understanding how to read and use these tables is essential in solving probability problems involving normal distributions.
Probability
Probability in the context of normal distributions refers to the likelihood that a given score or data point will fall within a certain area under the curve. This area represents the proportion of scores within a range and is expressed as a number between 0 and 1.
  • A probability less than 0.5 indicates that the event is less likely to occur, while a probability more than 0.5 indicates that it is likely to occur.
  • When working with a standard normal distribution, probability corresponds to the area under the curve for specific z-scores.
Calculating probability is important for making predictions and informed decisions based on statistical data. The z-table simplifies finding these probabilities by providing pre-calculated areas under the curve.
Area Under Curve
The area under the curve in a normal distribution is a concept that represents probability. In a standard normal distribution, the total area under the curve equals 1, indicating a 100% probability. Any specific section of this area (or curve) matches the likelihood of a statistic falling within that range.
  • If you want to find the likelihood of a data point being less than a certain value, you check the area to the left of that point.
  • Conversely, if you want the probability of a data point being more than a certain value, you consider the area to the right.
The transformation from raw scores to z-scores makes these calculations possible with the help of z-tables, as it standardizes the probabilities across all datasets. Understanding the area concept helps you comprehend the distribution and frequency of data in a more intuitive way.

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Most popular questions from this chapter

Alpha Corporation is considering two suppliers to secure the large amounts of steel rods that it uses. Company A produces rods with a mean diameter of \(8 \mathrm{~mm}\) and a standard deviation of \(.15 \mathrm{~mm}\) and sells 10,000 rods for \(\$ 400\). Company B produces rods with a mean diameter of \(8 \mathrm{~mm}\) and a standard deviation of \(.12 \mathrm{~mm}\) and sells 10,000 rods for \(\$ 460\). A rod is usable only if its diameter is between \(7.8 \mathrm{~mm}\) and \(8.2 \mathrm{~mm}\). Assume that the diameters of the rods produced by each company have a normal distribution. Which of the two companies should Alpha Corporation use as a supplier? Justify your answer with appropriate calculations.

Find the area under the standard normal curve a. from \(z=0\) to \(z=3.94\) b. between \(z=0\) and \(z=-5.16\) c. to the right of \(z=5.42\) d. to the left of \(z=-3.68\)

Let \(x\) be a continuous random variable that follows a normal distribution with a mean of 550 and a standard deviation of 75 . a. Find the value of \(x\) so that the area under the normal curve to the left of \(x\) is \(.0250\). b. Find the value of \(x\) so that the area under the normal curve to the right of \(x\) is \(.9345\). c. Find the value of \(x\) so that the area under the normal curve to the right of \(x\) is approximately \(.0275 .\) d. Find the value of \(x\) so that the area under the normal curve to the left of \(x\) is approximately . 9600 . e. Find the value of \(x\) so that the area under the normal curve between \(\mu\) and \(x\) is approximately \(.4700\) and the value of \(x\) is less than \(\mu .\) f. Find the value of \(x\) so that the area under the normal curve between \(\mu\) and \(x\) is approximately \(.4100\) and the value of \(x\) is greater than \(\mu\).

The print on the package of Sylvania CFL \(65 \mathrm{~W}\) replacement bulbs that use only \(16 \mathrm{~W}\) claims that these bulbs have an average life of 8000 hours. Assume that the distribution of lives of all such bulbs is normal with a mean of 8000 hours and a standard deviation of 400 hours. Let \(x\) be the life of a randomly selected such light bulb. a. Find \(x\) so that about \(22.5 \%\) of such light bulbs have lives longer than this value. b. Find \(x\) so that about \(63 \%\) of such light bulbs have lives shorter than this value.

Find the following binomial probabilities using the normal approximation. a. \(n=70, \quad p=.30, \quad P(x=18)\) b. \(n=200, \quad p=.70, \quad P(133 \leq x \leq 145)\) c. \(n=85, \quad p=.40, \quad P(x \geq 30)\) d. \(n=150, \quad p=.38, \quad P(x \leq 62)\)
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