6.1a Parameters of the Normal Distribution

Porter was told that his IQ score was 1 standard deviation below the mean. If IQ scores were approximately normal with μ=75 and σ=6, what was Porter's score? Do not include units in your answer. For example, if you found that the score was 75 points, you would enter 75.

69

Annie was told that her math test score was 3 standard deviations below the mean. If test scores were approximately normal with μ=99 and σ=4, what was Annie's score? Do not include units in your answer. For example, if you found that the score was 99 points, you would enter 99.

87

Horace was told that his math test score was 3 standard deviations above the mean. If test scores were approximately normal with μ=81 and σ=6, what was Horace's score? Do not include units in your answer. For example, if you found that the score was 81 points, you would enter 81.

99

The graph below shows the graphs of several normal distributions, labeled A, B, and C, on the same axis. Determine which normal distribution has the smallest mean.

A Remember that the mean of a normal distribution is the x-value of its central point (the top of the "hill"). Therefore, a distribution with a larger mean will be centered farther to the right than a distribution with a smaller mean.The distribution that is farthest to the left is A, so that has the smallest mean.

The graph below shows the graphs of several normal distributions, labeled A, B, and C, on the same axis. Determine which normal distribution has the largest mean.

A Remember that the mean of a normal distribution is the x-value of its central point (the top of the "hill"). Therefore, a distribution with a larger mean will be centered farther to the right than a distribution with a smaller mean.The distribution that is farthest to the right is A, so that has the largest mean.

Suppose X∼N(6.5,1.5), and x=3.5. Find and interpret the z-score of the standardized normal random variable.

The z-score when x=3.5 is −2. The mean is 6.5. This z-score tells you that x=3.5 is 2 standard deviations to the left of the mean.

Suppose X∼N(13.5,1.5), and x=9. Find and interpret the z-score of the standardized normal random variable.

The z-score when x=9 is −3. The mean is 13.5. This z-score tells you that x=9 is 3 standard deviations to the left of the mean.

The graph below shows the graphs of several normal distributions, labeled A, B, and C, on the same axis. Determine which normal distribution has the smallest standard deviation.

C Remember that the standard deviation tells how spread out the normal distribution is. So a high standard deviation means the graph will be short and spread out. A low standard deviation means the graph will be tall and skinny. The distribution that is the tallest and least spread out is C, so that has the smallest standard deviation.

John averages 58 words per minute on a typing test with a standard deviation of 11 words per minute. Suppose John's words per minute on a typing test are normally distributed. Let X= the number of words per minute on a typing test. Then X∼N(58,11). If necessary, round to three decimal places.

Suppose John types 72 words per minute in a typing test on Sunday. The z-score when x = 72 is 1.273. The mean is 58. This z-score tells you that x = 72 is 1.273 standard deviations to the right of the mean.

Rosetta averages 148 points per bowling game with a standard deviation of 14 points. Suppose Rosetta's points per bowling game are normally distributed. Let X= the number of points per bowling game. Then X∼N(148,14). If necessary, round to three decimal places.

Suppose Rosetta scores 110 points in the game on Thursday. The z-score when x=110 is -2.714. The mean is 148. This z-score tells you that x=110 is 2.714 standard deviations to the left of the mean.

Floretta's points per basketball game are normally distributed with a standard deviation of 4 points. If Floretta scores 10 points, and the z-score of this value is −4, then what is her mean points in a game?

26 Points

Given the plot of normal distributions A and B below, which of the following statements is true? Select all correct answers.

B has the larger mean. A has the larger standard deviation. Remember that the mean of a normal distribution is the x-value of its central point (the top of the "hill"). Therefore, a distribution with a larger mean will be centered farther to the right than a distribution with a smaller mean.Because B is farther to the right than A, the mean of B is greater than the mean of A.Remember that the standard deviation tells how spread out the normal distribution is. So a high standard deviation means the graph will be short and spread out. A low standard deviation means the graph will be tall and skinny.Because A is shorter and more spread out than B, we find that A has the larger standard deviation.

Given the plot of normal distributions A and B below, which of the following statements is true? Select all correct answers.

B has the larger mean. B has the larger standard deviation. Remember that the mean of a normal distribution is the x-value of its central point (the top of the "hill"). Therefore, a distribution with a larger mean will be centered farther to the right than a distribution with a smaller mean.Because B is farther to the right than A, the mean of B is greater than the mean of A.Remember that the standard deviation tells how spread out the normal distribution is. So a high standard deviation means the graph will be short and spread out. A low standard deviation means the graph will be tall and skinny.Because B is shorter and more spread out than A, we find that B has the larger standard deviation.

Given the plot of normal distributions A and B below, which of the following statements is true? Select all correct answers.

B has the larger mean. The standard deviations of A and B are equal. Remember that the mean of a normal distribution is the x-value of its central point (the top of the "hill"). Therefore, a distribution with a larger mean will be centered farther to the right than a distribution with a smaller mean.Because B is farther to the right than A, the mean of B is greater than the mean of A.Remember that the standard deviation tells how spread out the normal distribution is. So a high standard deviation means the graph will be short and spread out. A low standard deviation means the graph will be tall and skinny.Because the distributions are the same height and equally spread out, we find that the standard deviations of A and B are equal.

Given the plot of normal distributions A and B below, which of the following statements is true? Select all correct answers.

B has the larger mean. A has the larger standard deviation. Remember that the mean of a normal distribution is the x-value of its central point (the top of the "hill"). Therefore, a distribution with a larger mean will be centered farther to the right than a distribution with a smaller mean.Because B is farther to the right than A, the mean of B is greater than the mean of A.Remember that the standard deviation tells how spread out the normal distribution is. So a high standard deviation means the graph will be short and spread out. A low standard deviation means the graph will be tall and skinny.Because A is shorter and more spread out than B, we find that A has the larger standard deviation.

Given the plot of normal distributions A and B below, which of the following statements is true? Select all correct answers.

The means of A and B are equal. A has the larger standard deviation. Remember that the mean of a normal distribution is the x-value of its central point (the top of the "hill"). Therefore, a distribution with a larger mean will be centered farther to the right than a distribution with a smaller mean.Because A and B are centered at the same point, their means are equal.Remember that the standard deviation tells how spread out the normal distribution is. So a high standard deviation means the graph will be short and spread out. A low standard deviation means the graph will be tall and skinny.Because A is shorter and more spread out than B, we find that A has the larger standard deviation.

Given the plot of normal distributions A and B below, which of the following statements is true? Select all correct answers.

The means of A and B are equal. B has the larger standard deviation. Remember that the mean of a normal distribution is the x-value of its central point (the top of the "hill"). Therefore, a distribution with a larger mean will be centered farther to the right than a distribution with a smaller mean.Because A and B are centered at the same point, their means are equal.Remember that the standard deviation tells how spread out the normal distribution is. So a high standard deviation means the graph will be short and spread out. A low standard deviation means the graph will be tall and skinny.Because B is shorter and more spread out than A, we find that B has the larger standard deviation.