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Chapter 7: Problem 12

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 (a) to the left of \(Z=-2.94\) or to the right of \(Z=2.94\) (b) to the left of \(Z=-1.68\) or to the right of \(Z=3.05\) (c) to the left of \(Z=-0.88\) or to the right of \(Z=1.23\)

Short Answer

Expert verified
(a) 0.0032, (b) 0.0476, (c) 0.2987

Step by step solution

01

Identify Z-scores and corresponding areas

For each problem, identify the Z-scores given and the areas of interest under the standard normal curve. Use the Z-table to find the areas corresponding to the given Z-scores.
02

Convert Z-scores to areas using Z-table

1. For each Z-score, look up the cumulative area to the left of the Z-score in the Z-table. 2. For (a), look up the area to the left of -2.94 and the area to the right of 2.94. For standard normal distribution, the area to the right of a positive Z-score is equal to 1 minus the area to the left of the Z-score. Therefore, the areas of interest are: For Z = -2.94, the area to the left (as per Z-table) is 0.0016. For Z = 2.94, the area to the right is 1 - area to the left = 1 - 0.9984 = 0.0016. The total area for (a) is 0.0016 + 0.0016 = 0.0032. For (b): For Z = -1.68, the area to the left (as per Z-table) is 0.0465. For Z = 3.05, the area to the right is 1 - area to the left = 1 - 0.9989 = 0.0011. The total area for (b) is 0.0465 + 0.0011 = 0.0476. For (c): For Z = -0.88, the area to the left (as per Z-table) is 0.1894. For Z = 1.23, the area to the right is 1 - area to the left = 1 - 0.8907 = 0.1093. The total area for (c) is 0.1894 + 0.1093 = 0.2987.
03

Sum the areas for each problem

For each part (a), (b), and (c), sum the areas to the left and right of the respective Z-scores to get the total area.
04

Draw the standard normal curve

Draw a standard normal curve for each part and shade the areas corresponding to the Z-scores. The shaded regions should match the calculated areas.

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Key Concepts

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

Z-scores
Z-scores are a crucial part of understanding the standard normal curve. A Z-score measures how many standard deviations a data point is from the mean.
Z-scores can be positive or negative. A positive Z-score indicates the data point is above the mean, while a negative Z-score indicates it is below.
The formula to calculate a Z-score is:
\[ Z = \frac{(X - \mu)}{\sigma} \]
Where:
  • X is the value of the data point
  • \(\mu\) is the mean of the data set
  • \(\sigma\) is the standard deviation of the data set.

Using Z-scores, we can find the position of any data point in the context of the standard normal distribution. In the problem presented, each Z-score represents a position on the standard normal curve that we need to consider for calculating areas.
cumulative area
Cumulative area in the context of the standard normal distribution represents the total area under the curve to the left (or right) of a specific Z-score. This is essential for determining probabilities.
Each section under the standard normal curve represents a probability. Key probabilities in statistics are found by looking at the cumulative area to a specific Z-score.
For example, the cumulative area to the left of a Z-score indicates the probability of a data point being less than that specific value.
Let's look at the following steps:
  • For a given Z-score, find the cumulative area to the left. This is straightforward from the Z-table.
  • To find the cumulative area to the right of a Z-score, subtract the left cumulative area from 1 (since the total area under the curve is 1).
In the given exercise, cumulative areas are found for specific Z-scores to identify the required probabilities.
Z-table
The Z-table (also known as the standard normal table) lists the cumulative probabilities for each Z-score. It helps us determine the area under the standard normal curve up to a specific Z-score.
Here's how to use a Z-table:
  • Identify the Z-score for which you need the cumulative area.
  • Find the Z-score in the Z-table. Typically, the rows represent the first two digits, and the columns represent the second decimal place. For instance, a Z-score of -2.94 is found at the intersection of the row for -2.9 and the column for 0.04.
  • Read the corresponding cumulative area from the table. This value represents the area under the curve to the left of the Z-score.

In the exercise, the Z-table is used to find areas corresponding to Z-scores like -2.94 and 2.94.
area under the curve
The area under the standard normal curve represents probabilities for normally distributed data. The total area under the curve is always 1.
Each section of the curve between Z-scores corresponds to the probability of values within that range.
For negative Z-scores, the area to the left is directly found using the Z-table. For positive Z-scores, the area to the right is calculated using:
\[ P(Z > z_0) = 1 - P(Z \leq z_0) \]
Here are some key steps:
  • Identify the Z-score and find the cumulative area from the Z-table.
  • If looking for the area to the right, subtract the found cumulative area from 1.
  • Add areas if looking at combined regions (e.g., areas to the left of a negative Z-score, or to the right of its positive counterpart).
This process allows us to solve for the areas specified in the original exercise.

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Most popular questions from this chapter

Steel rods are manufactured with a mean length of 25 centimeter \((\mathrm{cm}) .\) Because of variability in the maufacturing process, the lengths of the rods are approximateIy normally distributed with a standard deviation of \(0.07 \mathrm{cm} .\) (a) What proportion of rods has a length less than 24.9 \(\mathrm{cm} ?\) (b) Any rods that are shorter than \(24.85 \mathrm{cm}\) or longer than \(25.15 \mathrm{cm}\) are discarded. What proportion of rods will be discarded? (c) Using the results of part (b), if 5000 rods are manufactured in a day, how many should the plant manager expect to discard? (d) If an order comes in for 10,000 steel rods, how many rods should the plant manager manufacture if the order states that all rods must be between \(24.9 \mathrm{cm}\) and \(25.1 \mathrm{cm} ?\)

Compute \(P(x)\) using the binomial probability formula. Then determine whether the normal distribution can be used as an approximation for the binomial distribution. If so, approximate \(P(x)\) and compare the result to the exact probability. $$n=75, p=0.75, X=60$$

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 \(0.98 .\)

The number of chocolate chips in an 18 -ounce bag of Chips Ahoy! chocolate chip cookies is approximately normally distributed with a mean of 1262 chips and standard deviation 118 chips according to a study by cadets of the U.S. Air Force Academy. (Source: Brad Warner and Jim Rutledge, Chance, Vol. 12, No. \(1,1999,\) pp. \(10-14 .\) ) (a) What is the probability that a randomly selected 18 ounce bag of Chips Ahoy! cookies contains between 1000 and 1400 chocolate chips? (b) What is the probability that a randomly selected 18 ounce bag of Chips Ahoy! cookies contains fewer than 1000 chocolate chips? (c) What proportion of 18 -ounce bags of Chip Ahoy! cookies contains more than 1200 chocolate chips? (d) What proportion of 18 -ounce bags of Chip Ahoy! cookies contains fewer than 1125 chocolate chips? (e) What is the percentile rank of an 18 -ounce bag of Chip Ahoy! cookies that contains 1475 chocolate chips? (f) What is the percentile rank of an 18-ounce bag of Chip Ahoy! cookies that contains 1050 chocolate chips?

Compute \(P(x)\) using the binomial probability formula. Then determine whether the normal distribution can be used as an approximation for the binomial distribution. If so, approximate \(P(x)\) and compare the result to the exact probability. $$n=40, p=0.25, X=30$$
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