Suppose A Normal Distribution Has A Mean Of 79 And A Standard Deviation Of 7. What Is $P(x \geq 65)$?A. 0.975 B. 0.16 C. 0.025 D. 0.84
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Understanding the Normal Distribution
A normal distribution, also known as a Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In this distribution, the mean, median, and mode are all equal. The normal distribution is characterized by its mean (μ) and standard deviation (σ).
Given Information
We are given a normal distribution with a mean (μ) of 79 and a standard deviation (σ) of 7. We need to find the probability that a value (x) is greater than or equal to 65, denoted as P(x ≥ 65).
Standardizing the Value
To find the probability, we need to standardize the value 65 by converting it into a z-score. The z-score formula is:
z = (x - μ) / σ
where x is the value we want to standardize, μ is the mean, and σ is the standard deviation.
Calculating the Z-Score
Using the given values, we can calculate the z-score as follows:
z = (65 - 79) / 7 = -14 / 7 = -2
Finding the Probability
Now that we have the z-score, we can use a standard normal distribution table (also known as a z-table) to find the probability that a value is greater than or equal to 65. The z-table shows the probability that a value is less than or equal to a given z-score.
Using the Z-Table
Looking up the z-score of -2 in the z-table, we find that the probability that a value is less than or equal to -2 is approximately 0.0228. Since we want to find the probability that a value is greater than or equal to 65, we need to subtract this value from 1.
Calculating the Final Probability
P(x ≥ 65) = 1 - P(x ≤ -2) = 1 - 0.0228 = 0.9772
Rounding this value to two decimal places, we get:
P(x ≥ 65) ≈ 0.98
However, this is not among the given options. We need to find the closest option.
Comparing with Given Options
Comparing our calculated value with the given options, we see that the closest option is:
A. 0.975
This is the correct answer.
Conclusion
In this article, we learned how to calculate probabilities in a normal distribution using the z-score formula and a standard normal distribution table. We applied this knowledge to find the probability that a value is greater than or equal to 65 in a normal distribution with a mean of 79 and a standard deviation of 7. The correct answer is:
A. 0.975
Additional Tips and Resources
- To find the probability that a value is less than or equal to a given z-score, use a standard normal distribution table (z-table).
- To find the probability that a value is greater than or equal to a given z-score, subtract the probability that a value is less than or equal to the z-score from 1.
- For more information on normal distributions and z-scores, see the resources listed below.
Resources
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Q: What is a normal distribution?
A: A normal distribution, also known as a Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
Q: What is a z-score?
A: A z-score is a measure of how many standard deviations an element is from the mean. It is calculated using the formula: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
Q: How do I calculate a z-score?
A: To calculate a z-score, you need to know the value (x), the mean (μ), and the standard deviation (σ). You can use the formula: z = (x - μ) / σ.
Q: What is the z-table?
A: The z-table, also known as a standard normal distribution table, is a table that shows the probability that a value is less than or equal to a given z-score.
Q: How do I use the z-table?
A: To use the z-table, look up the z-score in the table and find the corresponding probability. This probability is the chance that a value is less than or equal to the given z-score.
Q: How do I find the probability that a value is greater than or equal to a given z-score?
A: To find the probability that a value is greater than or equal to a given z-score, subtract the probability that a value is less than or equal to the z-score from 1.
Q: What is the difference between a z-score and a standard deviation?
A: A z-score is a measure of how many standard deviations an element is from the mean, while a standard deviation is a measure of the amount of variation or dispersion of a set of values.
Q: How do I calculate the standard deviation?
A: To calculate the standard deviation, you need to know the values of the data set. You can use the formula: σ = √[(Σ(x - μ)^2) / (n - 1)], where x is the value, μ is the mean, and n is the number of values.
Q: What is the mean?
A: The mean is the average of a set of values. It is calculated by adding up all the values and dividing by the number of values.
Q: How do I calculate the mean?
A: To calculate the mean, you need to know the values of the data set. You can use the formula: μ = (Σx) / n, where x is the value and n is the number of values.
Q: What is the median?
A: The median is the middle value of a set of values when they are arranged in order. If there is an even number of values, the median is the average of the two middle values.
Q: How do I calculate the median?
A: To calculate the median, you need to know the values of the data set. You can arrange the values in order and find the middle value.
Q: What is the mode?
A: The mode is the value that appears most frequently in a set of values.
Q: How do I calculate the mode?
A: To calculate the mode, you need to know the values of the data set. You can count the frequency of each value and find the value that appears most frequently.
Q: What is the range?
A: The range is the difference between the largest and smallest values in a set of values.
Q: How do I calculate the range?
A: To calculate the range, you need to know the values of the data set. You can find the largest and smallest values and subtract the smallest value from the largest value.
Q: What is the interquartile range (IQR)?
A: The interquartile range (IQR) is the difference between the 75th percentile and the 25th percentile of a set of values.
Q: How do I calculate the IQR?
A: To calculate the IQR, you need to know the values of the data set. You can arrange the values in order and find the 25th and 75th percentiles.
Q: What is the standard normal distribution?
A: The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
Q: How do I use the standard normal distribution?
A: To use the standard normal distribution, you can use the z-table to find the probability that a value is less than or equal to a given z-score.
Q: What is the difference between a normal distribution and a standard normal distribution?
A: A normal distribution is a probability distribution that is symmetric about the mean, while a standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
Q: How do I calculate the probability that a value is less than or equal to a given z-score?
A: To calculate the probability that a value is less than or equal to a given z-score, you can use the z-table.
Q: How do I calculate the probability that a value is greater than or equal to a given z-score?
A: To calculate the probability that a value is greater than or equal to a given z-score, you can subtract the probability that a value is less than or equal to the z-score from 1.
Q: What is the z-score formula?
A: The z-score formula is: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
Q: What is the standard deviation formula?
A: The standard deviation formula is: σ = √[(Σ(x - μ)^2) / (n - 1)], where x is the value, μ is the mean, and n is the number of values.
Q: What is the mean formula?
A: The mean formula is: μ = (Σx) / n, where x is the value and n is the number of values.
Q: What is the median formula?
A: The median formula is: median = (n + 1) / 2, where n is the number of values.
Q: What is the mode formula?
A: The mode formula is: mode = the value that appears most frequently.
Q: What is the range formula?
A: The range formula is: range = max(x) - min(x), where x is the value.
Q: What is the IQR formula?
A: The IQR formula is: IQR = Q3 - Q1, where Q3 is the 75th percentile and Q1 is the 25th percentile.
Q: What is the standard normal distribution formula?
A: The standard normal distribution formula is: N(0, 1), where 0 is the mean and 1 is the standard deviation.
Q: What is the difference between a normal distribution and a standard normal distribution?
A: A normal distribution is a probability distribution that is symmetric about the mean, while a standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
Q: How do I use the standard normal distribution?
A: To use the standard normal distribution, you can use the z-table to find the probability that a value is less than or equal to a given z-score.
Q: What is the z-table?
A: The z-table, also known as a standard normal distribution table, is a table that shows the probability that a value is less than or equal to a given z-score.
Q: How do I use the z-table?
A: To use the z-table, look up the z-score in the table and find the corresponding probability. This probability is the chance that a value is less than or equal to the given z-score.
Q: How do I find the probability that a value is greater than or equal to a given z-score?
A: To find the probability that a value is greater than or equal to a given z-score, subtract the probability that a value is less than or equal to the z-score from 1.
Q: What is the difference between a z-score and a standard deviation?
A: A z-score is a measure of how many standard deviations an element is from the mean, while a standard deviation is a measure of the amount of variation or dispersion of a set of values.
Q: How do I calculate the standard deviation?
A: To calculate the standard deviation, you need to know the values of the data set. You can use the formula: σ = √[(Σ(x - μ)^2) / (n - 1)], where x is the value, μ is the mean, and n is the number of values.
Q: What is the mean?
A: The mean is the average of a set of values. It is calculated by adding up all the values and dividing by