Section 1
Normal Distributions
What are the characteristics of a normal distribution?
Why is the standard normal distribution important in statistical analysis?
What is the total area under the standard normal distribution curve?
What percentage of the area falls below the mean? Above the mean?
About what percentage of the area under the normal distribution curve falls within 1 standard deviation above and below the mean? 2 standard deviations? 3 standard deviations?
What are two other names for a normal distribution?
Find the area under the standard normal distribution curve.Between z = 0 and z = 1.07
Find the area under the standard normal distribution curve.Between z = 0 and z = 1.77
Find the area under the standard normal distribution curve.Between z = 0 and z = 1.93
Find the area under the standard normal distribution curve.Between z = 0 and z = ?0.32
Find the area under the standard normal distribution curve.To the right of z = 0.37
Find the area under the standard normal distribution curve.To the right of z = 2.01
Find the area under the standard normal distribution curve.To the left of z = ?1.87
Find the area under the standard normal distribution curve.$$\text { To the left of } z=-0.75$$
Find the area under the standard normal distribution curve.Between z = 1.09 and z = 1.83
Find the area under the standard normal distribution curve.Between z = 1.23 and z = 1.90
Find the area under the standard normal distribution curve.$$\text { Between } z=-1.46 \text { and } z=-1.77$$
Find the area under the standard normal distribution curve.$$\text { Between } z=-0.96 \text { and } z=-0.36$$
Procter $\&$ Gamble reported that an American family of four washes an average of 1 ton ( 2000 pounds) of clothes each year. If the standard deviation of the distribution is 187.5 pounds, find the probability that the mean of a randomly selected sample of 50 families of four will be between 1980 and 1990 pounds.
Find the area under the standard normal distribution curve.Between z = 0.24 and z = ?1.12
Find the area under the standard normal distribution curve.To the left of z = 1.12
Find the area under the standard normal distribution curve.To the left of z = 1.31
Find the area under the standard normal distribution curve.To the right of z = ?0.18
Find the area under the standard normal distribution curve.To the right of z = 1.92?
Find the area under the standard normal distribution curveTo the right of $z=1.92$ and to the left of $z=-0.44$
Find the area under the standard normal distribution curve.To the left of $z=-2.15$ and to the right of $z=1.62$
Find the probabilities for each, using the standard normal distribution.$P(0< z<0.95)$
Find the probabilities for each, using the standard normal distribution.$P(0< z<1.96)$
Find the probabilities for each, using the standard normal distribution.$P(-1.38< z<0)$
Find the probabilities for each, using the standard normal distribution.$P(-1.23< z<0)$
Find the probabilities for each, using the standard normal distribution.$P(z>2.33)$
Find the probabilities for each, using the standard normal distribution.$P(z>0.82)$
Find the probabilities for each, using the standard normal distribution.$P(z< -1.51)$
Find the probabilities for each, using the standard normal distribution.$P(z< -1.77)$
Find the probabilities for each, using the standard normal distribution.$P(-2.07< z<1.88)$
Find the probabilities for each, using the standard normal distribution.$P(-0.20< z<1.56)$
Find the probabilities for each, using the standard normal distribution.$P(1.56< z<2.13)$
Find the probabilities for each, using the standard normal distribution.$P(1.12< z<1.43)$
Find the probabilities for each, using the standard normal distribution.$P(z< 1.42)$
Find the probabilities for each, using the standard normal distribution.$P(z>-1.43)$
Find the z value that corresponds to the given area.0.4175
Find the z value that corresponds to the given area.0.4066
Find the z value that corresponds to the given area.0.0188
Find the z value that corresponds to the given area.0.0239
Find the z value that corresponds to the given area.0.8962
Find the z value that corresponds to the given area.0.9671
Find the z value to the left of the mean so thata. 98.87% of the area under the distribution curve lies to the right of it.b. 82.12% of the area under the distribution curve lies to the right of it.c. 60.64% of the area under the distribution curve lies to the right of it.
Find the z value to the right of the mean so thata. 54.78% of the area under the distribution curve liesto the left of it.b. 69.85% of the area under the distribution curve liesto the left of it.c. 88.10% of the area under the distribution curve liesto the left of it.
Find two z values, one positive and one negative, that are equidistant from the mean so that the areas in the two tails add to the following values.a. 5%b. 10%c. 1%
Find two z values so that 48% of the middle area is bounded by them.
Find $P(-1< z<1), P(-2< z<2),$ and $P(-3< z<3)$How do these values compare with the empirical rule?
In the standard normal distribution, find the values of z for the 75th, 80th, and 92nd percentiles.
$z_{0}$ is the statistical notation for an unknown z value. It serves that same function as $x$ does in an algebraic equation.Find $z_{0}$ such that $P\left(-1.2<z<z_{0}\right)=0.8671$.
$z_{0}$ is the statistical notation for an unknown z value. It serves that same function as $x$ does in an algebraic equation.Find $z_{0}$ such that $P\left(z_{0}<z<2.5\right)=0.7672$.
$z_{0}$ is the statistical notation for an unknown z value. It serves that same function as $x$ does in an algebraic equation.Find $z_{2}$ such that the area between $z_{0}$ and $z=-0.5$ is 0.2345 (two answers).
$z_{0}$ is the statistical notation for an unknown z value. It serves that same function as $x$ does in an algebraic equation.Find $z_{0}$ such that $P\left(-z_{0}<z<z_{0}\right)=0.76$
Find the equation for the standard normal distribution by substituting 0 for $\mu$ and 1 for $\sigma$ in the equation$$y=\frac{e^{-(X-\mu)^{2} /\left(2 \sigma^{2}\right)}}{\sigma \sqrt{2 \pi}}$$
Graph by hand the standard normal distribution by using the formula derived in Exercise $57 .$ Let $\pi \approx 3.14$ and $e \approx 2.718 .$ Use $X$ values of $-2,-1.5,-1,-0.5,0,0.5,$ $1,1.5,$ and $2 .$ (Use a calculator to compute the $y$ values.)
Find $P(z<2.3 \text { or } z>-1.2)$
Find $P(z>2.3 \text { and } z<-1.2)$