6.E: The Standard Normal Distribution (Optional Exercises)
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Exercise \(\PageIndex{7}\)
A bottle of water contains 12.05 fluid ounces with a standard deviation of 0.01 ounces. Define the random variable \(X\) in words. \(X =\) ____________.
Answer
ounces of water in a bottle
Exercise \(\PageIndex{8}\)
A normal distribution has a mean of 61 and a standard deviation of 15. What is the median?
Exercise \(\PageIndex{9}\)
\(X \sim N(1, 2)\)
\(\sigma =\) _______
Answer
2
Exercise \(\PageIndex{10}\)
A company manufactures rubber balls. The mean diameter of a ball is 12 cm with a standard deviation of 0.2 cm. Define the random variable \(X\) in words. \(X =\) ______________.
Exercise \(\PageIndex{11}\)
\(X \sim N(-4, 1)\)
What is the median?
Answer
–4
Exercise \(\PageIndex{12}\)
\(X \sim N(3, 5)\)
\(\sigma =\) _______
Exercise \(\PageIndex{13}\)
\(X \sim N(-2, 1)\)
\(\mu =\) _______
Answer
–2
Exercise \(\PageIndex{14}\)
What does a \(z\)-score measure?
Exercise \(\PageIndex{15}\)
What does standardizing a normal distribution do to the mean?
Answer
The mean becomes zero.
Exercise \(\PageIndex{16}\)
Is \(X \sim N(0, 1)\) a standardized normal distribution? Why or why not?
Exercise \(\PageIndex{17}\)
What is the \(z\)-score of \(x = 12\), if it is two standard deviations to the right of the mean?
Answer
\(z = 2\)
Exercise \(\PageIndex{18}\)
What is the \(z\)-score of \(x = 9\), if it is 1.5 standard deviations to the left of the mean?
Exercise \(\PageIndex{19}\)
What is the \(z\)-score of \(x = -2\), if it is 2.78 standard deviations to the right of the mean?
Answer
\(z = 2.78\)
Exercise \(\PageIndex{20}\)
What is the \(z\)-score of \(x = 7\), if it is 0.133 standard deviations to the left of the mean?
Exercise \(\PageIndex{21}\)
Suppose \(X \sim N(2, 6)\). What value of x has a z -score of three?
Answer
\(x = 20\)
Exercise \(\PageIndex{22}\)
Suppose \(X \sim N(8, 1)\). What value of \(x\) has a \(z\)-score of –2.25?
Exercise \(\PageIndex{23}\)
Suppose \(X \sim N(9, 5)\). What value of \(x\) has a \(z\)-score of –0.5?
Answer
\(x = 6.5\)
Exercise \(\PageIndex{24}\)
Suppose \(X \sim N(2, 3)\). What value of \(x\) has a \(z\)-score of –0.67?
Exercise \(\PageIndex{25}\)
Suppose \(X \sim N(4, 2)\). What value of \(x\) is 1.5 standard deviations to the left of the mean?
Answer
\(x = 1\)
Exercise \(\PageIndex{26}\)
Suppose \(X \sim N(4, 2)\). What value of \(x\) is two standard deviations to the right of the mean?
Exercise \(\PageIndex{27}\)
Suppose \(X \sim N(8, 9)\). What value of \(x\) is 0.67 standard deviations to the left of the mean?
Answer
\(x = 1.97\)
Exercise \(\PageIndex{28}\)
Suppose \(X \sim N(-1, 12)\). What is the \(z\)-score of \(x = 2\)?
Exercise \(\PageIndex{29}\)
Suppose \(X \sim N(12, 6)\). What is the \(z\)-score of \(x = 2\)?
Answer
\(z = –1.67\)
Exercise \(\PageIndex{30}\)
Suppose \(X \sim N(9, 3)\). What is the \(z\)-score of \(x = 9\)?
Exercise \(\PageIndex{31}\)
Suppose a normal distribution has a mean of six and a standard deviation of 1.5. What is the \(z\)-score of \(x = 5.5\)?
Answer
\(z \approx –0.33\)
Exercise \(\PageIndex{32}\)
In a normal distribution, \(x = 5\) and \(z = –1.25\). This tells you that \(x = 5\) is ____ standard deviations to the ____ (right or left) of the mean.
Exercise \(\PageIndex{33}\)
In a normal distribution, \(x = 3\) and \(z = 0.67\). This tells you that \(x = 3\) is ____ standard deviations to the ____ (right or left) of the mean.
Answer
0.67, right
Exercise \(\PageIndex{34}\)
In a normal distribution, \(x = –2\) and \(z = 6\). This tells you that \(z = –2\) is ____ standard deviations to the ____ (right or left) of the mean.
Exercise \(\PageIndex{35}\)
In a normal distribution, \(x = –5\) and \(z = –3.14\). This tells you that \(x = –5\) is ____ standard deviations to the ____ (right or left) of the mean.
Answer
3.14, left
Exercise \(\PageIndex{36}\)
In a normal distribution, \(x = 6\) and \(z = –1.7\). This tells you that \(x = 6\) is ____ standard deviations to the ____ (right or left) of the mean.
Exercise \(\PageIndex{37}\)
About what percent of \(x\) values from a normal distribution lie within one standard deviation (left and right) of the mean of that distribution?
Answer
about 68%
Exercise \(\PageIndex{38}\)
About what percent of the \(x\) values from a normal distribution lie within two standard deviations (left and right) of the mean of that distribution?
Exercise \(\PageIndex{39}\)
About what percent of \(x\) values lie between the second and third standard deviations (both sides)?
Answer
about 4%
Exercise \(\PageIndex{40}\)
Suppose \(X \sim N(15, 3)\). Between what \(x\) values does 68.27% of the data lie? The range of \(x\) values is centered at the mean of the distribution (i.e., 15).
Exercise \(\PageIndex{41}\)
Suppose \(X \sim N(-3, 1)\). Between what \(x\) values does 95.45% of the data lie? The range of \(x\) values is centered at the mean of the distribution (i.e., –3).
Answer
between –5 and –1
Exercise \(\PageIndex{42}\)
Suppose \(X \sim N(-3, 1)\). Between what \(x\) values does 34.14% of the data lie?
Exercise \(\PageIndex{43}\)
About what percent of \(x\) values lie between the mean and three standard deviations?
Answer
about 50%
Exercise \(\PageIndex{44}\)
About what percent of \(x\) values lie between the mean and one standard deviation?
Exercise \(\PageIndex{45}\)
About what percent of \(x\) values lie between the first and second standard deviations from the mean (both sides)?
Answer
about 27%
Exercise \(\PageIndex{46}\)
About what percent of \(x\) values lie between the first and third standard deviations(both sides)?
Use the following information to answer the next two exercises: The life of Sunshine CD players is normally distributed with mean of 4.1 years and a standard deviation of 1.3 years. A CD player is guaranteed for three years. We are interested in the length of time a CD player lasts.
Exercise \(\PageIndex{47}\)
Define the random variable \(X\) in words. \(X =\) _______________.
Answer
The lifetime of a Sunshine CD player measured in years.
Exercise \(\PageIndex{48}\)
\(X \sim\) _____(_____,_____)