The normal distribution is perhaps the most frequently used distribution when discussing continuous random variables. It has several uses in statistics, some of which include its use in hypothesis testing, creating confidence intervals, and calculating p-values. Some variables that could be described using the normal distribution include the weights of various objects, classroom sizes, total snowfall amounts, and race times. The normal distribution has the following probability density function: f of x equals 1 divided by sigma times the square root of 2 pi, times e to the power of negative x minus mu, squared, divided by 2 sigma squared……..

Standardizing a Normally Distributed Random Variable

Standardize a normally distributed random variable

Question

In 2014, the CDC estimated that the mean height for adult women in the U.S. was 64 inches with a standard deviation of 4inches. Suppose X, height in inches of adult women, follows a normal distribution. Let x=68, the height of a woman who is 5' 8" tall. Find and interpret the z-score of the standardized normal random variable.

Select the correct answer below:

Understand the notation and interpret the parameters of a normal distribution

Question

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 .

Standardize a normally distributed random variable

Question

Suppose X∼N(18,2), and x=22. Find and interpret the z-score of the standardized normal random variable.

Understand the notation and interpret the parameters of a normal distribution

Question

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.

Understand the notation and interpret the parameters of a normal distribution

Question

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

Question

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

 Understand the notation and interpret the parameters of a normal distribution

Question

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.

Understand the notation and interpret the parameters of a normal distribution

Question

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

Understand the notation and interpret the parameters of a normal distribution

Parameters of the Normal Distribution

When introducing many tools for statistical inference, like hypothesis testing, confidence intervals, and P-values, we will start by looking at examples that use the Normal probability distribution. While these tools can use any distribution, the most common is the Normal probability distribution because it is a good approximation for many random variables measured in real-world applications.

Understand the notation and interpret the parameters of a normal distribution

Question

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 standard deviation.  

Understand the notation and interpret the parameters of a normal distribution

Question

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

Understand the notation and interpret the parameters of a normal distribution

Question

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 standard deviation.   

Understand the notation and interpret the parameters of a normal distribution

Question

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.

Understand the standard deviation of a set of data

Question

Which of the following lists of data has the smallest standard deviation?

Select the correct answer below:

Understand the standard deviation of a set of data

Question

Which of the following lists of data has the smallest standard deviation?

Select the correct answer below:

Understand the notation and interpret the parameters of a normal distribution

Question

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

Understand the notation and interpret the parameters of a normal distribution

Question

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

Understand the notation and interpret the parameters of a normal distribution

Question

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.  

Understand the notation and interpret the parameters of a normal distribution

Question

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

Standardize a normally distributed random variable

Question

Suppose X∼N(12.5,1.5), and x=11. Find and interpret the z-score of the standardized normal random variable.

Standardize a normally distributed random variable

Question

Isabella averages 17 points per basketball game with a standard deviation of 4 points. Suppose Isabella's points per basketball game are normally distributed. Let X= the number of points per basketball game. Then X∼N(17,4). Suppose Isabella scores 29 points in the game on Sunday. The z-score when x = 29 is 3. The mean is 17. This z-score tells you that x = 29 is 3 standard deviations to the  right of the mean.

Standardize a normally distributed random variable

Standardizing a Normally Distributed Random Variable

Question   

Suppose X∼N(17,6), and x=5. Find and interpret the z-score of the standardized normal random variable.

Example 2

Question

The mean height of teenage males from Chile is 170 cm with a standard deviation of 6.28 cm. Male heights are known to follow a normal distribution. Let X = the height of a teenage male from Chile. Then X∼N(170,6.28). Correctly fill in the blanks in the statement below.

Standardize a normally distributed random variable

Question

Suppose X∼N(5.5,2), and x=7.5. Find and interpret the z-score of the standardized normal random variable. Select the correct answer below:

Standardize a normally distributed random variable

Question

Jerome averages 16 points a game with a standard deviation of 4 points. Suppose Jerome's points per game are normally distributed. Let X = the number of points per game. Then X∼N(16,4). 

Standardize a normally distributed random variable

Question

Suppose X∼N(10,0.5), and x=11.5. Find and interpret the z-score of the standardized normal random variable. Provide your answer below:

Standardize a normally distributed random variable

Question

Annie averages 23 points per basketball game with a standard deviation of 4 points. Suppose Annie's points per basketball game are normally distributed. Let X= the number of points per basketball game. Then X∼N(23,4). Provide your answer below:

Standardize a normally distributed random variable

Question

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

Standardize a normally distributed random variable

Question

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

Standardize a normally distributed random variable

Question

In 2014, the CDC estimated that the mean height for adult women in the U.S. was 64 inches with a standard deviation of 4inches. Suppose X, height in inches of adult women, follows a normal distribution. Let x=68, the height of a woman who is 5' 8" tall. Find and interpret the z-score of the standardized normal random variable.

Standardize a normally distributed random variable

Question

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. Provide your answer below:

Standardize a normally distributed random variable

Question

Gail averages 64 words per minute on a typing test with a standard deviation of 9.5 words per minute. Suppose Gail'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(64,9.5).

Standardize a normally distributed random variable

Question

William averages 58 words per minute on a typing test with a standard deviation of 10.5 words per minute. Suppose William'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(58,10.5).  If necessary, round to three decimal places.

Standardize a normally distributed random variable

Question

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

Standardize a normally distributed random variable

Question

Suppose X∼N(16.5,0.5), and x=16. Find and interpret the z-score of the standardized normal random variable. Provide your answer below: