Question 1

State the Properties of the Normal Curve
-Approximately _____$\%$ of the area under the normal curve is between $\mu - 3 \sigma$ and $\mu + 3 \sigma$.

Accepted Answer

The property of the normal curve states that approximately 99.7% of the data falls within 3 standard deviations ($\sigma$) from the mean ($\mu$), which is known as the empirical rule or the 68-95-99.7 rule.

Question 2

State the Properties of the Normal Curve
-The normal density curve is symmetric about

Accepted Answer

The normal curve is symmetric about its mean, meaning that the left half of the curve is a mirror image of the right half. This means that the curve is not symmetric about the horizontal axis, an inflection point, or any other point located a certain distance from the mean.

Question 3

Provide an appropriate response
-A random number generator is set top generate integer random numbers between 1 and 10 inclusive following a uniform distribution. What is the probability of the random number generator generating a 7

Accepted Answer

A

Question 4

Provide an appropriate response
-A machine is set to pump cleanser into a process at the rate of 5 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 4.5 to 7.5 gallons per minute. Find the probability that between 5.0 gallons and 6.0 gallons are pumped during a randomly selected minute.

Accepted Answer

The answer of Provide an appropriate response
-A machine is set...

Question 5

Explain the Role of Area in the Normal Density Function
-The weight of 2-year old hyraxes is known to be normally distributed with mean $\mu = 2200$ grams and standard deviation $\sigma = 365$ grams
(a) Draw a normal curve with the parameters labeled.
(b) Shade the region that represents the proportion of hyraxes who weighed more than 2930 grams.
(c) Suppose the area under the normal curve to the left of $X = 2930$ is $0.0228$. Provide two interpretations of this result.

Accepted Answer

The answer of Explain the Role of Area in the...

Question 6

State the Properties of the Normal Curve
-The highest point on the graph of the normal density curve is located at

Accepted Answer

The highest point on a normal density curve corresponds to its mean ($\mu$), which is also the median and mode in a perfectly symmetrical distribution.

Question 7

Explain the Role of Area in the Normal Density Function
-The analytic scores on a standardized aptitude test are known to be normally distributed with mean $\mu = 610$ and standard deviation $\sigma = 115$.
(a) Draw a normal curve with the parameters labeled.
(b) Shade the region that represents the proportion of test takers who scored less than 725 .
(c) Suppose the area under the normal curve to the left of $X = 725$ is $0.8413$. Provide two interpretations of this result.

Accepted Answer

The answer of Explain the Role of Area in the...

Question 8

Provide an appropriate response
-Suppose x is a uniform random variable with c = 10 and d = 90. Find the probability that a randomly selected observation exceeds 26.

Accepted Answer

The probability that a randomly selected observation exceeds 26 can be found by calculating the area under the uniform distribution curve from 26 to 90. Since the total range is from 10 to 90, the length of the interval is 80. The interval from 26 to 90 has a length of 64. Therefore, the probability is 64/80 = 0.8.

Question 9

Provide an appropriate response
-High temperatures in a certain city for the month of August follow a uniform distribution over the interval 64°F to 94°F. Find the high temperature which 90% of the August days exceed.

Accepted Answer

The answer of Provide an appropriate response
-High temperatures in a...

Question 10

Provide an appropriate response
-Suppose a uniform random variable can be used to describe the outcome of an experiment with the outcomes ranging from 10 to 70. What is the probability that this experiment results in an outcome less than 20

Accepted Answer

The probability is calculated by taking the range of desired outcomes (10 to 20, which is 10 units) and dividing it by the total range of outcomes (10 to 70, which is 60 units). Therefore, 10/60 = 1/6 ≈ 0.17.

Question 11

State the Properties of the Normal Curve
-The graph of a normal curve is given. Use the graph to identify the value of $\mu$ and $\sigma$.

Accepted Answer

The answer of State the Properties of the Normal Curve
-The...

Question 12

Graph a Normal Curve
-Compare a graph of the normal density function with mean of 0 and standard deviation of 1 with a graph of a normal density function with mean equal to 4 and standard deviation of 1. The graphs would

Accepted Answer

The graphs would have the same shape and height, but one would be shifted 4 units to the right (since the mean is 4 units greater). The standard deviation does not affect the horizontal position of the curve, only its steepness.

Question 13

Provide an appropriate response
-Suppose x is a uniform random variable with c = 30 and d = 70. Find the probability that a randomly selected observation is between 33 and 65.

Accepted Answer

The probability that a randomly selected observation falls between two points in a uniform distribution is calculated as the difference between those points divided by the range of the distribution. Here, the range is $d - c = 70 - 30 = 40$, and the interval of interest is $65 - 33 = 32$. Thus, the probability is $32 / 40 = 0.8$.

Question 14

Provide an appropriate response
-In a uniform probability distribution, any random variable is just as likely as any other random variable to occur, provided the random variables belong to the distribution.

Accepted Answer

In a uniform probability distribution, every outcome has an equal chance of occurring, meaning any specific value within the distribution's range is just as likely to be observed as any other value within the same range.

Question 15

Provide an appropriate response
-High temperatures in a certain city for the month of August follow a uniform distribution over the interval 65°F to 87°F. What is the probability that a randomly selected August day has a high temperature that exceeded 70°F

Accepted Answer

The range of temperatures is 87°F - 65°F = 22°F. The range of temperatures exceeding 70°F is 87°F - 70°F = 17°F. The probability is the favorable range over the total range, so 17/22 = 0.7727.

Question 16

State the Properties of the Normal Curve
-The area under a standard normal density curve with mean of 0 and standard deviation of 1 is

Accepted Answer

The area under any normal density curve, including a standard normal curve (mean = 0, standard deviation = 1), is always 1. This is because the total area under the curve represents the total probability of all outcomes, which must sum to 1.

Question 17

Graph a Normal Curve
-Compare a graph of the normal density function with mean of 0 and standard deviation of 1 with a graph of a normal density function with mean equal to 0 and standard deviation of 0.5. The graphs would

Accepted Answer

The graphs would have no horizontal displacement since they share the same mean (0), but the graph with a smaller standard deviation (0.5) would be steeper compared to the graph with a standard deviation of 1.

Question 18

Provide an appropriate response
-The diameter of ball bearings produced in a manufacturing process can be explained using a uniform distribution over the interval 6.5 to 8.5 millimeters. What is the probability that a randomly selected ball bearing has a diameter greater than 7.4 millimeters

Accepted Answer

The uniform distribution is defined over the interval from 6.5 to 8.5 millimeters, making the total length of the interval 2 millimeters (8.5 - 6.5). The part of the interval greater than 7.4 millimeters extends from 7.4 to 8.5, which is 1.1 millimeters long (8.5 - 7.4). The probability is the length of the desired interval (1.1) divided by the total length of the interval (2), which equals 0.55.

Question 19

Explain the Role of Area in the Normal Density Function
-The average mpg (miles per gallon) of a new model of motorcycle is known to be normally distributed with mean $\mu = 27.4 \mathrm { mpg }$ and standard deviation $\sigma = 2.9 \mathrm { mpg }$.
(a) Draw a normal curve with the parameters labeled.
(b) Shade the region that represents the proportion of mpgs between $29.3$ and $23.7$.
(c) Suppose the area under the normal curve to between $X = 29.3$ and $X = 23.7$ is $0.6419$. Provide two interpretations of this result.

Accepted Answer

The answer of Explain the Role of Area in the...

Question 20

Graph a Normal Curve
-You are performing a study about the weight of preschoolers. A previous study found the weights to be normally distributed with a mean of 30 and a standard deviation of 4. You randomly sample 30 preschool children and find their weights to be as follows. $$\begin{array} { l l l l l l l l l l } 
25 & 25 & 26 & 26.5 & 27 & 27 & 27.5 & 28 & 28 & 28.5 \
29 & 29 & 30 & 30 & 30.5 & 31 & 31 & 32 & 32.5 & 32.5 \
33 & 33 & 34 & 34.5 & 35 & 35 & 37 & 37 & 38 & 38
\end{array}$$ a) Draw a histogram to display the data. Is it reasonable to assume that the weights are normally distributed Why b) Find the mean and standard deviation of your sample. c) Is there a high probability that the mean and standard deviation of your sample are consistent with those found in previous studies Explain your reasoning.

Accepted Answer

The answer of Graph a Normal Curve
-You are performing a...

Quiz 7: The Normal Probability Distribution

State the Properties of the Normal Curve -Approximately _____ $\%$ of the area under the normal curve is between $\mu - 3 \sigma$ and $\mu + 3 \sigma$ .

State the Properties of the Normal Curve -The normal density curve is symmetric about

Provide an appropriate response -A random number generator is set top generate integer random numbers between 1 and 10 inclusive following a uniform distribution. What is the probability of the random number generator generating a 7

State the Properties of the Normal Curve -The highest point on the graph of the normal density curve is located at

Provide an appropriate response -Suppose x is a uniform random variable with c = 10 and d = 90. Find the probability that a randomly selected observation exceeds 26.

Provide an appropriate response -High temperatures in a certain city for the month of August follow a uniform distribution over the interval 64°F to 94°F. Find the high temperature which 90% of the August days exceed.

Provide an appropriate response -Suppose a uniform random variable can be used to describe the outcome of an experiment with the outcomes ranging from 10 to 70. What is the probability that this experiment results in an outcome less than 20

Graph a Normal Curve -Compare a graph of the normal density function with mean of 0 and standard deviation of 1 with a graph of a normal density function with mean equal to 4 and standard deviation of 1. The graphs would

Provide an appropriate response -Suppose x is a uniform random variable with c = 30 and d = 70. Find the probability that a randomly selected observation is between 33 and 65.

Provide an appropriate response -In a uniform probability distribution, any random variable is just as likely as any other random variable to occur, provided the random variables belong to the distribution.

Provide an appropriate response -High temperatures in a certain city for the month of August follow a uniform distribution over the interval 65°F to 87°F. What is the probability that a randomly selected August day has a high temperature that exceeded 70°F

State the Properties of the Normal Curve -The area under a standard normal density curve with mean of 0 and standard deviation of 1 is

Graph a Normal Curve -Compare a graph of the normal density function with mean of 0 and standard deviation of 1 with a graph of a normal density function with mean equal to 0 and standard deviation of 0.5. The graphs would

Provide an appropriate response -The diameter of ball bearings produced in a manufacturing process can be explained using a uniform distribution over the interval 6.5 to 8.5 millimeters. What is the probability that a randomly selected ball bearing has a diameter greater than 7.4 millimeters

Quiz 7: The Normal Probability Distribution

State the Properties of the Normal Curve -Approximately _____ %\%% of the area under the normal curve is between μ−3σ\mu - 3 \sigmaμ−3σ and μ+3σ\mu + 3 \sigmaμ+3σ .

State the Properties of the Normal Curve -The normal density curve is symmetric about

Provide an appropriate response -A random number generator is set top generate integer random numbers between 1 and 10 inclusive following a uniform distribution. What is the probability of the random number generator generating a 7

State the Properties of the Normal Curve -The highest point on the graph of the normal density curve is located at

Provide an appropriate response -Suppose x is a uniform random variable with c = 10 and d = 90. Find the probability that a randomly selected observation exceeds 26.

Provide an appropriate response -High temperatures in a certain city for the month of August follow a uniform distribution over the interval 64°F to 94°F. Find the high temperature which 90% of the August days exceed.

Provide an appropriate response -Suppose a uniform random variable can be used to describe the outcome of an experiment with the outcomes ranging from 10 to 70. What is the probability that this experiment results in an outcome less than 20

State the Properties of the Normal Curve -The graph of a normal curve is given. Use the graph to identify the value of μ\muμ and σ\sigmaσ .

Graph a Normal Curve -Compare a graph of the normal density function with mean of 0 and standard deviation of 1 with a graph of a normal density function with mean equal to 4 and standard deviation of 1. The graphs would

Provide an appropriate response -Suppose x is a uniform random variable with c = 30 and d = 70. Find the probability that a randomly selected observation is between 33 and 65.

Provide an appropriate response -In a uniform probability distribution, any random variable is just as likely as any other random variable to occur, provided the random variables belong to the distribution.

Provide an appropriate response -High temperatures in a certain city for the month of August follow a uniform distribution over the interval 65°F to 87°F. What is the probability that a randomly selected August day has a high temperature that exceeded 70°F

State the Properties of the Normal Curve -The area under a standard normal density curve with mean of 0 and standard deviation of 1 is

Graph a Normal Curve -Compare a graph of the normal density function with mean of 0 and standard deviation of 1 with a graph of a normal density function with mean equal to 0 and standard deviation of 0.5. The graphs would

Provide an appropriate response -The diameter of ball bearings produced in a manufacturing process can be explained using a uniform distribution over the interval 6.5 to 8.5 millimeters. What is the probability that a randomly selected ball bearing has a diameter greater than 7.4 millimeters

State the Properties of the Normal Curve -Approximately _____ $\%$ of the area under the normal curve is between $\mu - 3 \sigma$ and $\mu + 3 \sigma$ .