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Chapter 7: Problem 10

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) \(z=-2.55\) and \(z=2.55\) (b) \(z=-1.67\) and \(z=0\) (c) \(z=-3.03\) and \(z=1.98\)

Short Answer

Expert verified
(a) 0.9892, (b) 0.4525, (c) 0.9749

Step by step solution

01

- Identify the Required Areas

First, identify the z-scores given in the problem for each part and note that we need to find the area under the standard normal curve between those z-scores.
02

- Understand the Standard Normal Curve

Remember that the standard normal curve is a bell-shaped curve symmetrical around the mean, which is zero. The total area under the curve is always 1.
03

- Use Z-Table for Area Calculation

Use the Z-table (standard normal distribution table) to find the area to the left of each z-score. The Z-table gives the area under the curve to the left of a given z-score.
04

- Calculate the Area for Part (a)

For part (a), find the area between \(z = -2.55\) and \(z = 2.55\).- From the Z-table, the area to the left of \(z = 2.55\) is approximately 0.9946.- The area to the left of \(z = -2.55\) is approximately 0.0054.- Subtract the areas: \(0.9946 - 0.0054 = 0.9892\).
05

- Calculate the Area for Part (b)

For part (b), find the area between \(z = -1.67\) and \(z = 0\).- From the Z-table, the area to the left of \(z = 0\) is 0.5.- The area to the left of \(z = -1.67\) is approximately 0.0475.- Subtract the areas: \(0.5 - 0.0475 = 0.4525\).
06

- Calculate the Area for Part (c)

For part (c), find the area between \(z = -3.03\) and \(z = 1.98\).- From the Z-table, the area to the left of \(z = 1.98\) is approximately 0.9761.- The area to the left of \(z = -3.03\) is approximately 0.0012.- Subtract the areas: \(0.9761 - 0.0012 = 0.9749\).

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Key Concepts

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

z-score
A z-score is a numerical measurement that describes the position of a value within a standard normal distribution. It tells us how many standard deviations a particular value is from the mean. If the z-score is positive, the value is above the mean; if it is negative, the value is below the mean. This score helps us understand the relative standing of the value within the overall data set. For example, a z-score of 2.55 means the value is 2.55 standard deviations above the mean.
Z-table
The Z-table, also known as the standard normal distribution table, provides the area under the standard normal curve to the left of a given z-score. This table is essential when calculating probabilities and comparing scores in a normal distribution. By using the Z-table, we can determine the proportion of values that fall below a specific z-score, which is important in various statistical analyses. For instance, to calculate the area between two z-scores, like in the exercise, we look up each z-score in the Z-table and subtract the smaller area from the larger one.
bell-shaped curve
The standard normal curve, also known as the bell-shaped curve, is a symmetrical graph that represents the distribution of data points where most values cluster around the mean. The curve is highest at the mean and tapers off towards the extremes, reflecting fewer occurrences of values far from the mean. This characteristic shape helps in understanding how data is spread around the average. The total area under the bell-shaped curve is always 1, which signifies 100% of the probability.
area under the curve
The area under the standard normal curve corresponds to the probabilities or proportions of the distribution. For instance, if we want to find the probability of a value falling between two z-scores, we calculate the area under the curve between these z-scores. In the provided exercise, finding the area under the curve between z = -2.55 and z = 2.55 involves using the Z-table to find the areas to the left of each z-score and subtracting these areas to get the final area, which represents the probability. This method applies to any pair of z-scores and helps in determining the likelihood of values within a specific range.

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Most popular questions from this chapter

In games where a team is favored by more than 12 points, the margin of victory for the favored team relative to the spread is normally distributed with a mean of -1.0 point and a standard deviation of 10.9 points. Source: Justin Wolfers, "Point Shaving: Corruption in NCAA Basketball' (a) In games where a team is favored by more than 12 points, what is the probability that the favored team wins by 5 or more points relative to the spread? (b) In games where a team is favored by more than 12 points, what is the probability that the favored team loses by 2 or more points relative to the spread? (c) In games where a team is favored by more than 12 points, what is the probability that the favored team "beats the spread"? Does this imply that the possible point shaving spreads are accurate for games in which a team is favored by more than 12 points?

The reading speed of sixth-grade students is approximately normal, with a mean speed of 125 words per minute and a standard deviation of 24 words per minute. (a) What is the reading speed of a sixth-grader whose reading speed is at the 90 th percentile? (b) A school psychologist wants to determine reading rates for unusual students (both slow and fast). Determine the reading rates of the middle \(95 \%\) of all sixth-grade students. What are the cutoff points for unusual readers?

Use a normal probability plot to assess whether the sample data could have come from a population that is normally distributed. O-Ring Thickness A random sample of O-rings was obtained, and the wall thickness (in inches) of each was recorded. $$\begin{array}{lllll} \hline 0.276 & 0.274 & 0.275 & 0.274 & 0.277 \\ \hline 0.273 & 0.276 & 0.276 & 0.279 & 0.274 \\ \hline 0.273 & 0.277 & 0.275 & 0.277 & 0.277 \\ \hline 0.276 & 0.277 & 0.278 & 0.275 & 0.276 \\ \hline \end{array}$$

Monthly charges for cell phone plans in the United States are normally distributed with mean \(\mu=\$ 62\) and standard deviation \(\sigma=\$ 18 .\) (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of plans that charge less than \(\$ 44\) (c) Suppose the area under the normal curve to the left of \(x=\$ 44\) is 0.1587 . Provide two interpretations of this result.

According to a USA Today "Snapshot," \(3 \%\) of Americans surveyed lie frequently. You conduct a survey of 500 college students and find that 20 of them lie frequently. (a) Compute the probability that, in a random sample of 500 college students, at least 20 lie frequently, assuming the true percentage is \(3 \%\) (b) Does this result contradict the USA Today "Snapshot"? Explain.
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