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The Standard Normal Distribution (Exercises)

  • Page ID
    5857
  • 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 \(\PageIndex{48}\)

    \(X \sim\) _____(_____,_____)