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8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305270336

Chapter 8.5, Problem 19E

Textbook Problem

For any normal distribution, find the probability that the random variable lies within two standard deviations of the mean.

Expert Solution

To determine

**To find:** The probability that the random variable lies within two standard deviations of the mean.

**Given information:**

The function is normal distribution.

**Calculation:**

Show the probability density function for normal distribution as follows.

Consider the standard means for normal distributions are

Find the probability density that the random variable lies within two standard deviations of the mean

Consider

Differentiate both sides of the Equation.

Calculate the lower limit of *t* using Equation (3).

Substitute
*x* in Equation (3)

Single Variable Calculus: Early Transcendentals

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Ch. 8.1 - Use the arc length formula (3) to find the length...Ch. 8.1 - Use the arc length formula to find the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Find the exact length of the curve. 9. y = 1 +...Ch. 8.1 - Find the exact length of the curve. 10. 36y2 = (x2...

Ch. 8.1 - Find the exact length of the curve. 11....Ch. 8.1 - Find the exact length of the curve. 12....Ch. 8.1 - Find the exact length of the curve. 13....Ch. 8.1 - Find the exact length of the curve. 14. y = ln(cos...Ch. 8.1 - Find the exact length of the curve. 15. y = ln(sec...Ch. 8.1 - Find the exact length of the curve. 16....Ch. 8.1 - Find the exact length of the curve. 17....Ch. 8.1 - Find the exact length of the curve. 18....Ch. 8.1 - Find the exact length of the curve. 19....Ch. 8.1 - Find the exact length of the curve. 20. y = 1 ex,...Ch. 8.1 - Find the length of the arc of the curve from point...Ch. 8.1 - Find the length of the arc of the curve from point...Ch. 8.1 - Graph the curve and visually estimate its length....Ch. 8.1 - Graph the curve and visually estimate its length....Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 8.1 - (a) Graph the curve y=x4x3,0x4. (b) Compute the...Ch. 8.1 - Repeat Exercise 29 for the curve y=x+sinx0x2Ch. 8.1 - Sketch the curve with equation x2/3 + y2/3 = 1 and...Ch. 8.1 - (a) Sketch the curve y3 = x2. (b) Use Formulas 3...Ch. 8.1 - Find the arc length function for the curve y =...Ch. 8.1 - (a) Find the arc length function for the curve y =...Ch. 8.1 - Find the arc length function for the curve...Ch. 8.1 - The arc length function for a curve y = f(x),...Ch. 8.1 - For the function f(x)=14ex+ex, prove that the arc...Ch. 8.1 - A steady wind blows a kite due west. The kites...Ch. 8.1 - A hawk flying at 15 m/s at an altitude of 180 m...Ch. 8.1 - The Gateway Arch in St. Louis (see the photo on...Ch. 8.1 - A manufacturer of corrugated metal roofing wants...Ch. 8.1 - (a) The figure shows a telephone wire hanging...Ch. 8.1 - Find the length of the curve y=1xt31dt1x4Ch. 8.1 - The curves with equations x + y = l , n = 4, 6, 8,...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use Simpsons Rule with n = 10 to approximate the...Ch. 8.2 - Use either a CAS or a table of integrals to find...Ch. 8.2 - Use either a CAS or a table of integrals to find...Ch. 8.2 - If the region R=(x,y)x1,0y1/x is rotated about the...Ch. 8.2 - If the infinite curve y = ex, x 0, is rotated...Ch. 8.2 - (a) If a 0, find the area of the surface...Ch. 8.2 - A group of engineers is building a parabolic...Ch. 8.2 - (a) The ellipsex2a2+y2b2=1abis rotated about the...Ch. 8.2 - Find the surface area of the torus in Exercise...Ch. 8.2 - If the curve y = f(x), a x b, is rotated about...Ch. 8.2 - Find the area of the surface obtained by rotating...Ch. 8.2 - (a) Show that the surface area of a zone of a...Ch. 8.2 - Show that if we rotate the curve y = ex/2 + ex/2...Ch. 8.2 - Let L be the length of the curve y = f(x), a x ...Ch. 8.2 - Formula 4 is valid only when f(x) 0. Show that...Ch. 8.3 - An aquarium 5 ft long, 2 ft wide, and 3 ft deep is...Ch. 8.3 - A tank is 8 m long, 4 m wide, 2 m high, and...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A milk truck carries milk with density 64.6 lb/ft3...Ch. 8.3 - A trough is filled with a liquid of density 840...Ch. 8.3 - A vertical dam has a semicircular gate as shown in...Ch. 8.3 - A cube with 20-cm-long sides is sitting on the...Ch. 8.3 - A dam is inclined at an angle of 30 from the...Ch. 8.3 - A swimming pool is 20 ft wide and 40 ft long and...Ch. 8.3 - Suppose that a plate is immersed vertically in a...Ch. 8.3 - A metal plate was found submerged vertically in...Ch. 8.3 - (a) Use the formula of Exercise 18 to show that...Ch. 8.3 - Point-masses mi are located on the x-axis as...Ch. 8.3 - Point-masses mi are located on the x-axis as...Ch. 8.3 - The masses mi are located at the points Pi. Find...Ch. 8.3 - The masses mi are located at the points Pi. Find...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Calculate the moments Mx and My and the center of...Ch. 8.3 - Calculate the moments Mx and My and the center of...Ch. 8.3 - Use Simpsons Rule to estimate the centroid of the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Use a graph to find approximate x-coordinates of...Ch. 8.3 - Prove that the centroid of any triangle is located...Ch. 8.3 - Find the centroid of the region shown, not by...Ch. 8.3 - Find the centroid of the region shown, not by...Ch. 8.3 - A rectangle with sides a and b is divided into two...Ch. 8.3 - If x is the x-coordinate of the centroid of the...Ch. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - The centroid of a curve can be found by a process...Ch. 8.3 - The Second Theorem of Pappus is in the same spirit...Ch. 8.3 - Use the Second Theorem of Pappus described in...Ch. 8.3 - Let R be the region that lies between the curves...Ch. 8.3 - Prove Formulas 9.Ch. 8.4 - The marginal cost function C(x) was defined to be...Ch. 8.4 - A company estimates that the marginal revenue (in...Ch. 8.4 - A mining company estimates that the marginal cost...Ch. 8.4 - The demand function for a particular vacation...Ch. 8.4 - A demand curve is given by p = 450/(x + 8). Find...Ch. 8.4 - The supply function ps(x) for a commodity gives...Ch. 8.4 - If a supply curve is modeled by the equation p =...Ch. 8.4 - In a purely competitive market, the price of a...Ch. 8.4 - The sum of consumer surplus and producer surplus...Ch. 8.4 - A camera company estimates that the demand...Ch. 8.4 - A company modeled the demand curve for its product...Ch. 8.4 - A movie theater has been charging 10.00 per person...Ch. 8.4 - If the amount of capital that a company has at...Ch. 8.4 - If revenue flows into a company at a rate of...Ch. 8.4 - If income is continuously collected at a rate of...Ch. 8.4 - The present value of an income stream is the...Ch. 8.4 - Paretos Law of Income states that the number of...Ch. 8.4 - A hot, wet summer is causing a mosquito population...Ch. 8.4 - Use Poiseuilles Law to calculate the rate of flow...Ch. 8.4 - High blood pressure results from constriction of...Ch. 8.4 - The dye dilution method is used to measure cardiac...Ch. 8.4 - After a 5.5-mg injection of dye, the readings of...Ch. 8.4 - The graph of the concentration function c(t) is...Ch. 8.5 - Let f (x) be the probability density function for...Ch. 8.5 - Let f(t) be the probability density function for...Ch. 8.5 - Let f (x) = 30x2(1 x)2 for 0 x 1 and f (x) = 0...Ch. 8.5 - The density function f(x)=e3x(1+e3x)2 is an...Ch. 8.5 - Let f (x) = c/(1 + x2). (a) For what value of c is...Ch. 8.5 - Let f(x) = k (3x x2) if 0 x 3 and f(x) = 0 if x...Ch. 8.5 - A spinner from a board game randomly indicates a...Ch. 8.5 - (a) Explain why the function whose graph is shown...Ch. 8.5 - Show that the median waiting time for a phone call...Ch. 8.5 - (a) A type of light bulb is labeled as having an...Ch. 8.5 - An online retailer has determined that the average...Ch. 8.5 - The time between infection and the display of...Ch. 8.5 - REM sleep is the phase of sleep when most active...Ch. 8.5 - According to the National Health Survey, the...Ch. 8.5 - The Garbage Project at the University of Arizona...Ch. 8.5 - Boxes are labeled as containing 500 g of cereal....Ch. 8.5 - The speeds of vehicles on a highway with speed...Ch. 8.5 - Show that the probability density function for a...Ch. 8.5 - For any normal distribution, find the probability...Ch. 8.5 - The standard deviation for a random variable with...Ch. 8.5 - The hydrogen atom is composed of one proton in the...Ch. 8 - (a) How is the length of a curve defined? (b)...Ch. 8 - (a) Write an expression for the surface area of...Ch. 8 - Describe how we can find the hydrostatic force...Ch. 8 - (a) What is the physical significance of the...Ch. 8 - What does the Theorem of Pappus say?Ch. 8 - Given a demand function p(x), explain what is...Ch. 8 - (a) What is the cardiac output of the heart? (b)...Ch. 8 - What is a probability density function? What...Ch. 8 - Suppose f(x) is the probability density function...Ch. 8 - What is a normal distribution? What is the...Ch. 8 - Find the length of the curve. 1. y = 4 (x 1)3/2,...Ch. 8 - Find the length of the curve. 2.y=2ln(sin12x), /3 ...Ch. 8 - Find the length of the curve. 3. 12x = 4y3 + 3y1,...Ch. 8 - (a) Find the length of the curve y=x416+12x21x2...Ch. 8 - Let C be the arc of the curve y = 2/(x + 1) from...Ch. 8 - (a) The curve y = x2, 0 x 1, is rotated about...Ch. 8 - Use Simpsons Rule with n = 10 to estimate the...Ch. 8 - Use Simpsons Rule with n = 10 to estimate the area...Ch. 8 - Find the length of the curve y=1xt1dt1x16Ch. 8 - Find the area of the surface obtained by rotating...Ch. 8 - A gate in an irrigation canal is constructed in...Ch. 8 - A trough is filled with water and its vertical...Ch. 8 - Find the centroid of the region shown. 13.Ch. 8 - Find the centroid of the region shown. 14.Ch. 8 - Find the centroid of the region bounded by the...Ch. 8 - Find the centroid of the region bounded by the...Ch. 8 - Find the volume obtained when the circle of radius...Ch. 8 - Use the Theorem of Pappus and the fact that the...Ch. 8 - The demand function for a commodity is given by...Ch. 8 - After a 6-mg injection of dye into a heart, the...Ch. 8 - (a) Explain why the function...Ch. 8 - Lengths of human pregnancies are normally...Ch. 8 - The length of time spent waiting in line at a...Ch. 8 - Find the area of the region S = {(x, y) | x 0, y ...Ch. 8 - Find the centroid of the region enclosed by the...Ch. 8 - If a sphere of radius r is sliced by a plane whose...Ch. 8 - (a) Show that an observer at height H above the...Ch. 8 - Suppose that the density of seawater, = (z),...Ch. 8 - The figure shows a semicircle with radius 1,...Ch. 8 - Let P be a pyramid with a square base of side 2b...Ch. 8 - Consider a flat metal plate to be placed...Ch. 8 - A uniform disk with radius 1 m is to be cut by a...Ch. 8 - A triangle with area 30 cm2 is cut from a corner...Ch. 8 - In a famous 18th-century problem, known as Buffons...Ch. 8 - If the needle in Problem 11 has length h L, its...Ch. 8 - Find the centroid of the region enclosed by the...

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