For A Standard Normal Distribution, Find The Approximate Value Of P ( − 0.78 ≤ Z ≤ 1.16 P(-0.78 \leq Z \leq 1.16 P ( − 0.78 ≤ Z ≤ 1.16 ]. Use The Portion Of The Standard Normal Table Below To Help Answer The Question. \[ \begin{tabular}{|c|c|} \hline Z$ & Probability \ \hline

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Understanding the Standard Normal Distribution

The standard normal distribution, also known as the z-distribution, is a type of normal distribution with a mean of 0 and a standard deviation of 1. It is a fundamental concept in statistics and is used to model a wide range of phenomena in various fields, including finance, engineering, and social sciences. The standard normal distribution is characterized by its bell-shaped curve, which is symmetric about the mean and has a total area of 1 under the curve.

Using the Standard Normal Table

To find the approximate value of P(−0.78≤z≤1.16)P(-0.78 \leq z \leq 1.16), we can use the portion of the standard normal table provided below.

zz Probability
-3.09 0.0008
-2.58 0.0049
-2.33 0.0100
-2.05 0.0199
-1.80 0.0359
-1.64 0.0500
-1.47 0.0633
-1.28 0.1003
-1.13 0.1293
-1.00 0.1587
-0.84 0.1999
-0.78 0.2181
-0.67 0.2489
-0.56 0.2863
-0.44 0.3271
-0.33 0.3707
-0.22 0.4140
-0.11 0.4599
0.00 0.5000
0.11 0.5401
0.22 0.5799
0.33 0.6197
0.44 0.6593
0.56 0.6987
0.67 0.7381
0.78 0.7761
0.84 0.8139
1.00 0.8413
1.13 0.8703
1.28 0.8993
1.47 0.9273
1.64 0.9500
1.80 0.9641
2.05 0.9779
2.33 0.9900
2.58 0.9949
3.09 0.9992

Finding the Probability

To find the approximate value of P(−0.78≤z≤1.16)P(-0.78 \leq z \leq 1.16), we need to find the probabilities corresponding to z=−0.78z = -0.78 and z=1.16z = 1.16 in the standard normal table.

From the table, we can see that the probability corresponding to z=−0.78z = -0.78 is 0.2181, and the probability corresponding to z=1.16z = 1.16 is 0.8767.

Calculating the Approximate Value

To find the approximate value of P(−0.78≤z≤1.16)P(-0.78 \leq z \leq 1.16), we can use the following formula:

P(−0.78≤z≤1.16)=P(z≤1.16)−P(z≤−0.78)P(-0.78 \leq z \leq 1.16) = P(z \leq 1.16) - P(z \leq -0.78)

Substituting the values from the table, we get:

P(−0.78≤z≤1.16)=0.8767−0.2181=0.6586P(-0.78 \leq z \leq 1.16) = 0.8767 - 0.2181 = 0.6586

Conclusion

In this article, we used the standard normal table to find the approximate value of P(−0.78≤z≤1.16)P(-0.78 \leq z \leq 1.16). We first found the probabilities corresponding to z=−0.78z = -0.78 and z=1.16z = 1.16 in the table, and then used the formula to calculate the approximate value of the probability. The approximate value of P(−0.78≤z≤1.16)P(-0.78 \leq z \leq 1.16) is 0.6586.

Understanding the Standard Normal Distribution

The standard normal distribution is a type of normal distribution with a mean of 0 and a standard deviation of 1. It is a fundamental concept in statistics and is used to model a wide range of phenomena in various fields, including finance, engineering, and social sciences. The standard normal distribution is characterized by its bell-shaped curve, which is symmetric about the mean and has a total area of 1 under the curve.

Using the Standard Normal Table

To find the approximate value of P(−0.78≤z≤1.16)P(-0.78 \leq z \leq 1.16), we can use the portion of the standard normal table provided below.

zz Probability
-3.09 0.0008
-2.58 0.0049
-2.33 0.0100
-2.05 0.0199
-1.80 0.0359
-1.64 0.0500
-1.47 0.0633
-1.28 0.1003
-1.13 0.1293
-1.00 0.1587
-0.84 0.1999
-0.78 0.2181
-0.67 0.2489
-0.56 0.2863
-0.44 0.3271
-0.33 0.3707
-0.22 0.4140
-0.11 0.4599
0.00 0.5000
0.11 0.5401
0.22 0.5799
0.33 0.6197
0.44 0.6593
0.56 0.6987
0.67 0.7381
0.78 0.7761
0.84 0.8139
1.00 0.8413
1.13 0.8703
1.28 0.8993
1.47 0.9273
1.64 0.9500
1.80 0.9641
2.05 0.9779
2.33 0.9900
2.58 0.9949
3.09 0.9992

Finding the Probability

To find the approximate value of P(−0.78≤z≤1.16)P(-0.78 \leq z \leq 1.16), we need to find the probabilities corresponding to z=−0.78z = -0.78 and z=1.16z = 1.16 in the standard normal table.

From the table, we can see that the probability corresponding to z=−0.78z = -0.78 is 0.2181, and the probability corresponding to z=1.16z = 1.16 is 0.8767.

Calculating the Approximate Value

To find the approximate value of P(−0.78≤z≤1.16)P(-0.78 \leq z \leq 1.16), we can use the following formula:

P(−0.78≤z≤1.16)=P(z≤1.16)−P(z≤−0.78)P(-0.78 \leq z \leq 1.16) = P(z \leq 1.16) - P(z \leq -0.78)

Substituting the values from the table, we get:

P(−0.78≤z≤1.16)=0.8767−0.2181=0.6586P(-0.78 \leq z \leq 1.16) = 0.8767 - 0.2181 = 0.6586

Conclusion

In this article, we used the standard normal table to find the approximate value of P(−0.78≤z≤1.16)P(-0.78 \leq z \leq 1.16). We first found the probabilities corresponding to z=−0.78z = -0.78 and z=1.16z = 1.16 in the table, and then used the formula to calculate the approximate value of the probability. The approximate value of P(−0.78≤z≤1.16)P(-0.78 \leq z \leq 1.16) is 0.6586.

Understanding the Importance of the Standard Normal Distribution

The standard normal distribution is a fundamental concept in statistics and is used to model a wide range of phenomena in various fields, including finance, engineering, and social sciences. It is characterized by its bell-shaped curve, which is symmetric about the mean and has a total area of 1 under the curve.

Using the Standard Normal Table

To find the approximate value of P(−0.78≤z≤1.16)P(-0.78 \leq z \leq 1.16), we can use the portion of the standard normal table provided below.

zz Probability
-3.09 0.

Q: What is the standard normal distribution?

A: The standard normal distribution, also known as the z-distribution, is a type of normal distribution with a mean of 0 and a standard deviation of 1. It is a fundamental concept in statistics and is used to model a wide range of phenomena in various fields, including finance, engineering, and social sciences.

Q: What is the bell-shaped curve of the standard normal distribution?

A: The bell-shaped curve of the standard normal distribution is a symmetric curve that is centered at the mean (0) and has a total area of 1 under the curve. The curve is characterized by its smooth, continuous shape and its symmetrical properties.

Q: How is the standard normal distribution used in statistics?

A: The standard normal distribution is used in statistics to model a wide range of phenomena, including the distribution of scores on a test, the distribution of heights of a population, and the distribution of stock prices. It is also used to calculate probabilities and to make inferences about a population based on a sample.

Q: How do I use the standard normal table to find probabilities?

A: To use the standard normal table to find probabilities, you need to find the probability corresponding to the z-score you are interested in. The z-score is the number of standard deviations away from the mean that a value is. You can use the table to find the probability that a value is less than or equal to a certain z-score.

Q: What is the difference between a z-score and a probability?

A: A z-score is a measure of how many standard deviations away from the mean a value is, while a probability is a measure of the likelihood of a value occurring. The z-score is used to find the probability, and the probability is used to make inferences about a population.

Q: How do I calculate the approximate value of a probability using the standard normal table?

A: To calculate the approximate value of a probability using the standard normal table, you need to find the probabilities corresponding to the z-scores you are interested in. You can then use the formula P(z ≤ z1) - P(z ≤ z2) to calculate the approximate value of the probability.

Q: What is the importance of the standard normal distribution in real-world applications?

A: The standard normal distribution is used in a wide range of real-world applications, including finance, engineering, and social sciences. It is used to model the distribution of stock prices, the distribution of heights of a population, and the distribution of scores on a test. It is also used to make inferences about a population based on a sample.

Q: How do I use the standard normal distribution to make inferences about a population?

A: To use the standard normal distribution to make inferences about a population, you need to calculate the z-score of a value and then use the standard normal table to find the probability that a value is less than or equal to that z-score. You can then use the probability to make inferences about the population.

Q: What are some common applications of the standard normal distribution?

A: Some common applications of the standard normal distribution include:

  • Modeling the distribution of stock prices
  • Modeling the distribution of heights of a population
  • Modeling the distribution of scores on a test
  • Making inferences about a population based on a sample
  • Calculating probabilities and making decisions based on those probabilities

Q: How do I choose the right z-score for a given problem?

A: To choose the right z-score for a given problem, you need to consider the mean and standard deviation of the data. You can then use the formula z = (X - μ) / σ to calculate the z-score, where X is the value, μ is the mean, and σ is the standard deviation.

Q: What are some common mistakes to avoid when using the standard normal distribution?

A: Some common mistakes to avoid when using the standard normal distribution include:

  • Not using the correct z-score
  • Not using the correct probability
  • Not considering the mean and standard deviation of the data
  • Not using the standard normal table correctly

Q: How do I use the standard normal distribution to solve problems in statistics?

A: To use the standard normal distribution to solve problems in statistics, you need to follow these steps:

  1. Define the problem and identify the z-score you need to find.
  2. Calculate the z-score using the formula z = (X - μ) / σ.
  3. Use the standard normal table to find the probability corresponding to the z-score.
  4. Use the probability to make inferences about the population.

Q: What are some resources for learning more about the standard normal distribution?

A: Some resources for learning more about the standard normal distribution include:

  • Textbooks on statistics and probability
  • Online tutorials and videos
  • Practice problems and exercises
  • Real-world applications and case studies

Q: How do I apply the standard normal distribution to real-world problems?

A: To apply the standard normal distribution to real-world problems, you need to follow these steps:

  1. Identify the problem and the data you need to analyze.
  2. Calculate the z-score using the formula z = (X - μ) / σ.
  3. Use the standard normal table to find the probability corresponding to the z-score.
  4. Use the probability to make inferences about the population and solve the problem.

Q: What are some common challenges when using the standard normal distribution?

A: Some common challenges when using the standard normal distribution include:

  • Not using the correct z-score
  • Not using the correct probability
  • Not considering the mean and standard deviation of the data
  • Not using the standard normal table correctly

Q: How do I overcome these challenges?

A: To overcome these challenges, you need to:

  • Double-check your calculations and use the correct z-score and probability.
  • Consider the mean and standard deviation of the data and use them correctly.
  • Use the standard normal table correctly and follow the instructions.
  • Practice using the standard normal distribution and apply it to real-world problems.