6.2.1: The Standard Normal Distribution (Exercises) (2024)

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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 \(\PageIndex{48}\)

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

    6.2.1: The Standard Normal Distribution (Exercises) (2024)

    FAQs

    How to solve standard normal distribution? ›

    Any point (x) from a normal distribution can be converted to the standard normal distribution (z) with the formula z = (x-mean) / standard deviation. z for any particular x value shows how many standard deviations x is away from the mean for all x values.

    How to find standard normal distribution on TI 84 Plus? ›

    The Normal Probability Distribution menu for the TI-84+ is found under DISTR (2nd VARS). NOTE: A mean of zero and a standard deviation of one are considered to be the default values for a normal distribution on the calculator, if you choose not to set these values.

    What are the values of the mean and standard deviation after converting to z scores? ›

    The z-score is the standardized value of random variable data, and mean and standard deviation of z-score value is equal to 0 and 1. The area for left sided value of z-score value can be obtained from the standard normal distribution table.

    What is the z-score for the standard normal distribution? ›

    A z-score of 0 indicates that the given point is identical to the mean. On the graph of the standard normal distribution, z = 0 is therefore the center of the curve. A positive z-value indicates that the point lies to the right of the mean, and a negative z-value indicates that the point lies left of the mean.

    What is the formula for calculating the normal distribution? ›

    z = (X – μ) / σ

    In probability theory, the normal or Gaussian distribution is a very common continuous probability distribution.

    How do you find the normal distribution in statistics? ›

    In order to be considered a normal distribution, a data set (when graphed) must follow a bell-shaped symmetrical curve centered around the mean. It must also adhere to the empirical rule that indicates the percentage of the data set that falls within (plus or minus) 1, 2 and 3 standard deviations of the mean.

    How to calculate z-score? ›

    The formula for calculating a z-score is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation. As the formula shows, the z-score is simply the raw score minus the population mean, divided by the population standard deviation.

    How do you find the standard distribution? ›

    1. Step 1: Find the mean.
    2. Step 2: Subtract the mean from each score.
    3. Step 3: Square each deviation.
    4. Step 4: Add the squared deviations.
    5. Step 5: Divide the sum by the number of scores.
    6. Step 6: Take the square root of the result from Step 5.

    What is an example of a normal distribution? ›

    A normal distribution is a common probability distribution . It has a shape often referred to as a "bell curve." Many everyday data sets typically follow a normal distribution: for example, the heights of adult humans, the scores on a test given to a large class, errors in measurements.

    How to find z-score without sample size? ›

    If you know the mean and standard deviation, you can find the z-score using the formula z = (x - μ) / σ where x is your data point, μ is the mean, and σ is the standard deviation.

    What is the z-score for dummies? ›

    Z-score is a result of standardizing an individual data point. Simply put, a z-score gives us an idea of how far the data point is from the mean measured in terms of standard deviation(σ). For instance, a z-score of 2.5 indicates that the value is between 2 to 3 standard deviations from the mean and is not so common.

    What is the symbol for standard deviation? ›

    Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lowercase Greek letter σ (sigma), for the population standard deviation, or the Latin letter s, for the sample standard deviation.

    What percent of data is within 2 standard deviations? ›

    Under this rule, 68% of the data falls within one standard deviation, 95% percent within two standard deviations, and 99.7% within three standard deviations from the mean.

    How to find p value from z? ›

    How do I find p-value from z-score?
    1. Left-tailed z-test: p-value = Φ(Zscore)
    2. Right-tailed z-test: p-value = 1 - Φ(Zscore)
    3. Two-tailed z-test: p-value = 2 × Φ(−|Zscore|) or. p-value = 2 - 2 × Φ(|Zscore|)
    Jan 18, 2024

    How do you find the SD of a normal distribution? ›

    You can calculate it by subtracting each data point from the mean value and then finding the squared mean of the differenced values; this is called Variance. The square root of the variance gives you the standard deviation.

    What is the formula for the normal distribution table? ›

    A normal distribution is defined by the following formula: f ( x ) = 1 σ 2 π e − 1 2 ( x − μ σ ) 2 , where: f(x) is the probability density function. x is a random variable.

    How to find z-score with mean and standard deviation? ›

    How do you find the z-score with mean and standard deviation? If you know the mean and standard deviation, you can find the z-score using the formula z = (x - μ) / σ where x is your data point, μ is the mean, and σ is the standard deviation.

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