Practice
6.1 The Standard Normal Distribution
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 = ____________.
A normal distribution has a mean of 61 and a standard deviation of 15. What is the median?
X ~ N(1, 2)
σ = _______
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 = ______________.
X ~ N(–4, 1)
What is the median?
X ~ N(3, 5)
σ = _______
X ~ N(–2, 1)
μ = _______
What does a z-score measure?
What does standardizing a normal distribution do to the mean?
Is X ~ N(0, 1) a standardized normal distribution? Why or why not?
What is the z-score of x = 12, if it is two standard deviations to the right of the mean?
What is the z-score of x = 9, if it is 1.5 standard deviations to the left of the mean?
What is the z-score of x = –2, if it is 2.78 standard deviations to the right of the mean?
What is the z-score of x = 7, if it is 0.133 standard deviations to the left of the mean?
Suppose X ~ N(2, 6). What value of x has a z-score of three?
Suppose X ~ N(8, 1). What value of x has a z-score of –2.25?
Suppose X ~ N(9, 5). What value of x has a z-score of –0.5?
Suppose X ~ N(2, 3). What value of x has a z-score of –0.67?
Suppose X ~ N(4, 2). What value of x is 1.5 standard deviations to the left of the mean?
Suppose X ~ N(4, 2). What value of x is two standard deviations to the right of the mean?
Suppose X ~ N(8, 9). What value of x is 0.67 standard deviations to the left of the mean?
Suppose X ~ N(–1, 2). What is the z-score of x = 2?
Suppose X ~ N(12, 6). What is the z-score of x = 2?
Suppose X ~ N(9, 3). What is the z-score of x = 9?
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?
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.
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.
In a normal distribution, x = –2 and z = 6. This tells you that x = –2 is ____ standard deviations to the ____ (right or left) of the mean.
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.
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.
About what percent of x values from a normal distribution lie within one standard deviation, left and right, of the mean of that distribution?
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?
About what percent of x values lie between the second and third standard deviations, both sides?
Suppose X ~ N(15, 3). Between what x values does 68.27 percent of the data lie? The range of x values is centered at the mean of the distribution (i.e., 15).
Suppose X ~ N(–3, 1). Between what x values does 95.45 percent of the data lie? The range of x values is centered at the mean of the distribution (i.e., –3).
Suppose X ~ N(–3, 1). Between what x values does 34.14 percent of the data lie?
About what percent of x values lie between the mean and three standard deviations?
About what percent of x values lie between the mean and one standard deviation?
About what percent of x values lie between the first and second standard deviations from the mean, both sides?
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.
Define the random variable X in words. X = _______________.
X ~ _____(_____, _____)
6.2 Using the Normal Distribution
How would you represent the area to the left of one in a probability statement?
What is the area to the right of one?
Is P(x 1) equal to P(x ≤ 1)? Why or why not?
How would you represent the area to the left of three in a probability statement?
What is the area to the right of three?
If the area to the left of x in a normal distribution is 0.123, what is the area to the right of x?
If the area to the right of x in a normal distribution is 0.543, what is the area to the left of x?
Use the following information to answer the next four exercises:
X ~ N(54, 8)
Find the probability that x > 56.
Find the probability that x 30.
Find the 80th percentile.
Find the 60th percentile.
X ~ N(6, 2)
Find the probability that x is between three and nine.
X ~ N(–3, 4)
Find the probability that x is between one and four.
X ~ N(4, 5)
Find the maximum of x in the bottom quartile.
Use the following information to answer the next three exercises: The life of Sunshine CD players is normally distributed with a 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. Find the probability that a CD player will break down during the guarantee period.
- Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.
- P(0 x ____________) = ___________. Use zero for the minimum value of x.
Find the probability that a CD player will last between 2.8 and 6 years.
- Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.
- P(__________ x __________) = __________
Find the 70th percentile of the distribution for the time a CD player lasts.
- Sketch the situation. Label and scale the axes. Shade the region corresponding to the lower 70 percent.
- P(x k) = __________. Therefore, k = _________.