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Given a normal distribution with \( \mu = 50 \) and \( \sigma = 5 \), and given you select a sample of \( n = 100 \), complete parts (a) through (d).

a. What is the probability that \( X \) is less than 49?
\[ P(X < 49) = 0.0228 \]
(Type an integer or decimal rounded to four decimal places as needed.)

b. What is the probability that \( X \) is between 49 and 51.5?
\[ P(49 < X < 51.5) = 0.9759 \]
(Type an integer or decimal rounded to four decimal places as needed.)

c. What is the probability that \( X \) is above 50.4?
\[ P(X > 50.4) = 0.2119 \]
(Type an integer or decimal rounded to four decimal places as needed.)

d. There is a 30% chance that \( X \) is above what value?
\[ X = \]
(Type an integer or decimal rounded to two decimal places as needed.)

Answer :

(a) The probability that X is less than 49 is 0.4207.

(b) The probability that X is between 49 and 51.5 is 0.1972.

(c) The probability that X is above 50.4 is 0.4681

(d) There is a 30% chance that X is above approximately 47.4.

Let’s break down each part:

(a) Probability that X is less than 49:

We have a normal distribution with a mean (μ) of 50 and a standard deviation (σ) of 5. We want to find P(X < 49).

Using the standard normal distribution (where Z is the z-score):

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

Substitute the given values:

[tex]Z = \frac{49 - 50}{5} = -0.2[/tex]

Now, we find the cumulative probability using the standard normal table or a calculator:

[tex]P(X < 49) = P(Z < -0.2) = 0.4207[/tex]

(b) Probability that X is between 49 and 51.5:

We want to find P(49 < X < 51.5)

First, calculate the z-scores for both values:

For 49: [tex]Z_1 = \frac{49 - 50}{5} = -0.2[/tex]

For 51.5: [tex]Z_2 = \frac{51.5 - 50}{5} = 0.3[/tex]

Now, find the cumulative probabilities:

[tex]P(49 < X < 51.5) = P(-0.2 < Z < 0.3) = 0.1972[/tex]

(c) Probability that X is above 50.4:

We want to find P(X > 50.4).

Calculate the z-score for 50.4:

[tex]Z = \frac{50.4 - 50}{5} = 0.08[/tex]

Find the cumulative probability:

[tex]P(X > 50.4) = P(Z > 0.08) = 0.4681[/tex]

(d) Value above which there is a 30% chance:

We need to find the z-score corresponding to the 30th percentile (0.30). Using the standard normal table or a calculator, we find:

[tex]Z_{0.30} \approx -0.52[/tex]

Now, solve for X:

[tex]X = \mu + Z_{0.30} \cdot \sigma = 50 + (-0.52) \cdot 5 = 47.4[/tex]

Therefore, there is a 30% chance that X is above approximately 47.4.

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Rewritten by : Barada

To find the probabilities, we use the standard normal distribution table or a calculator. P(X < 49) = 0.4207. P(49 < X < 51.5) = 0.1972. P(X > 50.4) = 0.4672. X > 47.38 with a 30% chance.

To find the probabilities, we need to use the standard normal distribution table or a calculator.

a. To find P(X < 49), we calculate the z-score using the formula z = (X - mu) / sigma, where X is 49, mu is 50, and sigma is 5.

Plugging in the values, we get z = (49 - 50) / 5 = -0.2. Using the z-table, we find that the probability for z = -0.2 is 0.4207.

Therefore, P(X < 49) = 0.4207.

b. To find P(49 < X < 51.5), we calculate the z-scores for both 49 and 51.5 and find the difference.

Using the z-table, we find that the z-score for 49 is -0.2 (same as part a) and the z-score for 51.5 is 0.3.

The probability for z = -0.2 is 0.4207, and the probability for z = 0.3 is 0.6179.

Taking the difference, we get P(49 < X < 51.5) = 0.6179 - 0.4207 = 0.1972.

c. To find P(X > 50.4), we calculate the z-score for 50.4 using the formula z = (X - mu) / sigma. Plugging in the values, we get z = (50.4 - 50) / 5 = 0.08. Using the z-table, we find that the probability for z = 0.08 is 0.5328.

Therefore, P(X > 50.4) = 1 - 0.5328 = 0.4672.

d. If there is a 30% chance that X is above a certain value, we need to find the z-score for that probability.

Using the z-table, we find that the z-score for a 30% probability is approximately -0.524.

Therefore, we can calculate the value of X by rearranging the formula: z = (X - mu) / sigma. Solving for X, we get X = z * sigma + mu = -0.524 * 5 + 50 = 47.38 (rounded to two decimal places).

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