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Chapter 7: Problem 5

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 to the left of (a) \(z=-2.45\) (b) \(z=-0.43\) (c) \(z=1.35\) (d) \(z=3.49\)

Short Answer

Expert verified
Areas to the left of: (a) 0.0071, (b) 0.3336, (c) 0.9115, (d) 0.9998

Step by step solution

01

- Overview of the Standard Normal Curve

The standard normal curve, also known as the Z-curve, is a bell-shaped curve that is symmetrical about the mean (0). It has a standard deviation of 1. Areas under this curve correspond to probabilities.
02

- Using the Standard Normal Table

To find the area under the curve to the left of a given z-value, refer to the standard normal (Z) table. This table provides the cumulative area from the left end of the curve up to the specified z-value.
03

Step 3a - Area to the Left of z = -2.45

Using the standard normal table, find the cumulative area that corresponds to z = -2.45. According to the table, the area is 0.0071. Therefore, the area to the left of z = -2.45 is 0.0071.
04

Step 3b - Area to the Left of z = -0.43

Using the standard normal table, find the cumulative area for z = -0.43. The table indicates an area of 0.3336. Hence, the area to the left of z = -0.43 is 0.3336.
05

Step 3c - Area to the Left of z = 1.35

Using the standard normal table, determine the cumulative area for z = 1.35. This corresponds to an area of 0.9115. Therefore, the area to the left of z = 1.35 is 0.9115.
06

Step 3d - Area to the Left of z = 3.49

From the standard normal table, find the cumulative area for z = 3.49. The area is approximately 0.9998. Thus, the area to the left of z = 3.49 is 0.9998.

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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 specific type of normal distribution that has a mean of 0 and a standard deviation of 1. Also known as the Z-distribution, it is a symmetrical, bell-shaped curve. Because it is standardized, it's easy to find probabilities for any given z-score using the standard normal table.

This table, sometimes called the Z-table, shows the cumulative probabilities from the left end up to a z-value. It's essential for solving problems related to the standard normal curve as it directly gives the area under the curve.
Z-Score
A z-score, or standard score, indicates how many standard deviations a specific data point is from the mean. A positive z-score means the data point is above the mean, while a negative z-score shows it's below the mean.

The z-score formula is:
\( Z = \frac{{X - \text{{mean}}}}{{\text{{standard deviation}}}} \)

In our exercise, we saw z-scores of -2.45, -0.43, 1.35, and 3.49. Each score points to a specific area under the standard normal curve, using the Z-table for exact values.
Cumulative Area
Cumulative area in the context of the standard normal distribution is the total area under the curve to the left of a given z-score. This area represents the probability that a randomly selected score from the distribution falls within this range.

For our exercise:
  • The area to the left of z = -2.45 is 0.0071
  • The area for z = -0.43 is 0.3336
  • The area for z = 1.35 is 0.9115
  • The area for z = 3.49 is 0.9998
Each value was found using the standard normal table, illustrating how cumulative areas facilitate understanding probabilities within the distribution.

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Most popular questions from this chapter

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Assume that the random variable \(X\) is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7 .\) Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. \(P(38

The mean incubation time of fertilized chicken eggs kept at \(100.5^{\circ} \mathrm{F}\) in a still-air incubator is 21 days. Suppose that 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.
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