High School

We appreciate your visit to Given a normal distribution with tex mu 50 tex and tex sigma 4 tex and given you select a sample of tex n 100 tex. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

Given a normal distribution with [tex]\mu=50[/tex] and [tex]\sigma=4[/tex], and given you select a sample of [tex]n=100[/tex], complete parts (a) through (d).

a. What is the probability that [tex]X[/tex] is less than 49?
[tex]P(X < 49) = [/tex] (Type an integer or decimal rounded to four decimal places as needed.)

b. What is the probability that [tex]\bar{X}[/tex] is between 49 and 51.5?
[tex]P(49 < \bar{X} < 51.5) = [/tex] (Type an integer or decimal rounded to four decimal places as needed.)

c. What is the probability that [tex]X[/tex] is above 50.1?
[tex]P(X > 50.1) = [/tex] (Type an integer or decimal rounded to four decimal places as needed.)

d. There is a 30% chance that [tex]\bar{X}[/tex] is above what value?
[tex]X = [/tex]

Answer :

a) Probability P(X < 49) is 0.4013. b) Probability P(49 < X- < 51.5) is 0.9938. c) Probability P(X > 50.1) is 0.4905. d) There is a 30% chance that X- is above approximately 52.0976.

To solve the given problems, we can use the properties of the normal distribution. Given a normal distribution with u = 50 and s = 4, and a sample size of n = 100, we can proceed as follows:

a. To find the probability that X is less than 49, we can use the cumulative distribution function (CDF) of the normal distribution. We want to calculate P(X < 49). Using the z-score formula, we can standardize the value of 49:

z = (x - u) / s

z = (49 - 50) / 4

z = -0.25

Using a standard normal distribution table or a calculator, we can find the corresponding cumulative probability for z = -0.25. Let's denote this probability as P(Z < -0.25).

P(X < 49) = P(Z < -0.25)

By looking up the value in the standard normal distribution table or using a calculator, we find that P(Z < -0.25) is approximately 0.4013.

Therefore, P(X < 49) ≈ 0.4013.

b. To find the probability that X- is between 49 and 51.5, we need to calculate P(49 < X- < 51.5). Since the sample size is large (n = 100), the sampling distribution of the sample mean will be approximately normally distributed. The mean of the sampling distribution is equal to the population mean (u = 50), and the standard deviation of the sampling distribution is equal to the population standard deviation divided by the square root of the sample size (s/√n = 4/√100 = 0.4).

We can now standardize the values of 49 and 51.5 using the sample mean distribution:

z1 = (x1 - u) / (s/√n) = (49 - 50) / 0.4 = -2.5

z2 = (x2 - u) / (s/√n) = (51.5 - 50) / 0.4 = 3.75

Now, we can find the probability P(49 < X- < 51.5) by subtracting the cumulative probabilities:

P(49 < X- < 51.5) = P(Z < 3.75) - P(Z < -2.5)

Using a standard normal distribution table or a calculator, we find that P(Z < 3.75) is approximately 1 and P(Z < -2.5) is approximately 0.0062.

Therefore, P(49 < X- < 51.5) ≈ 1 - 0.0062 = 0.9938.

c. To find the probability that X is above 50.1, we can use the CDF of the normal distribution. We want to calculate P(X > 50.1). Standardizing the value of 50.1:

z = (x - u) / s

z = (50.1 - 50) / 4

z = 0.025

The probability P(X > 50.1) is equal to 1 minus the cumulative probability P(X < 50.1) (from the standard normal distribution table or calculator):

P(X > 50.1) = 1 - P(Z < 0.025)

By looking up the value in the standard normal distribution table or using a calculator, we find that P(Z < 0.025) is approximately 0.5095.

Therefore, P(X > 50.1) ≈ 1 - 0.5095 = 0.4905.

d. To find the value of X such that there is a 30% chance that X- is above this value, we need to find the corresponding z-score from the standard normal distribution.

Let z be the z-score for which P(Z > z) = 0.3. From the standard normal distribution table or using a calculator, we find that P(Z > 0.5244) ≈ 0.3. Therefore, z ≈ 0.5244.

Now, we can use the formula for z-score to find the corresponding value of X:

z = (x - u) / s

Substituting the given values, we have:

0.5244 = (x - 50) / 4

Solving for x:

x - 50 = 0.5244 * 4

x - 50 = 2.0976

x ≈ 52.0976

Therefore, there is a 30% chance that X- is above approximately 52.0976.

To learn more about Probability here:

https://brainly.com/question/31828911

#SPJ4

Thanks for taking the time to read Given a normal distribution with tex mu 50 tex and tex sigma 4 tex and given you select a sample of tex n 100 tex. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada