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Given a normal distribution with μ=50 and σ=4, and given you select a sample of n=100, complete parts (a) through (d). a. What is the probability that X

ˉ

is less than 49 ? P( X

<49)= (Type an integer or decimal rounded to four decimal places as needed.) b. What is the probability that X

ˉ

is between 49 and 51.5 ? P(49< X

<51.5)= (Type an integer or decimal rounded to four decimal places as needed.) c. What is the probability that X

is above 50.3 ? P( X

ˉ

>50.3)= (Type an integer or decimal rounded to four decimal places as needed.) d. There is a 35% chance that X

ˉ

is above what value? X

= (Type an integer or decimal rounded to two decimal places as needed.)

Answer :

Final answer:

To solve this problem, we use the Central Limit Theorem. We find the z-scores and use them to find the probabilities.

Explanation:

To solve this problem, we can use the Central Limit Theorem. Since the sample size is large (n=100) and the population distribution is normal, the sample mean will also be approximately normally distributed.

a. To find the probability that X-bar is less than 49, we need to find the z-score for 49 using the formula:
z = (X - μ) / (σ / sqrt(n))
Then, we can find the probability using a standard normal distribution table or a calculator.

b. To find the probability that X-bar is between 49 and 51.5, we need to find the z-scores for both values and then find the corresponding probabilities for each z-score.

c. To find the probability that X-bar is above 50.3, we need to find the z-score for 50.3 and find the corresponding probability.

d. To find the value of X-bar that has a 35% chance of being above it, we need to find the z-score that corresponds to a probability of 0.35 and then find the corresponding value of X using the formula:
X = μ + (z * (σ / sqrt(n)))

Learn more about Probability here:

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