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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

General Electric manufactures a decorative Crystal Clear 60 -watt light bulb that it advertises will last 1500 hours. Suppose that the lifetimes of the light bulbs are approximately normally distributed, with a mean of 1550 hours and a standard deviation of 57 hours. (a) What proportion of the light bulbs will last less than the advertised time? (b) What proportion of the light bulbs will last more than 1650 hours? (c) What is the probability that a randomly selected GE Crystal Clear 60 -watt light bulb will last between 1625 and 1725 hours? (d) What is the probability that a randomly selected GE Crystal Clear 60 -watt light bulb will last longer than 1400 hours?

The lives of refrigerators are normally distributed with mean \(\mu=14\) years and standard deviation \(\sigma=2.5\) years Source: Based on information from Consumer Reports (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of refrigerators that last for more than 17 years. (c) Suppose the area under the normal curve to the right of \(x=17\) is 0.1151 . Provide two interpretations of this result.

True or False: A normal score is the expected z-score of a data value, assuming the distribution of the random variable is normal.

The birth weights of full-term babies are normally distributed with mean \(\mu=3400\) grams and \(\sigma=505\) grams. Source: Based on data obtained from the National Vital Statistics Report, Vol. \(48,\) No. 3 (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of full-term babies who weigh more than 4410 grams. (c) Suppose the area under the normal curve to the right of \(x=4410\) is \(0.0228 .\) Provide two interpretations of this result.

Use a normal probability plot to assess whether the sample data could have come from a population that is normally distributed. Customer Service A random sample of weekly work logs at an automobile repair station was obtained, and the average number of customers per day was recorded. $$\begin{array}{lllll} \hline 26 & 24 & 22 & 25 & 23 \\ \hline 24 & 25 & 23 & 25 & 22 \\ \hline 21 & 26 & 24 & 23 & 24 \\ \hline 25 & 24 & 25 & 24 & 25 \\ \hline 26 & 21 & 22 & 24 & 24 \\ \hline \end{array}$$
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