A random variable has a normal distribution with a mean of µ = 50 and the standard deviation σ = 5.2. What are the probabilities that the random variable will take on a value a. less than 55.2 b. greater than 60.3 c. between 52 and 57.2 d. between 40 and 65

To solve these probability questions for a normally distributed random variable \( X \) with mean \( \mu = 50 \) and standard deviation \( \sigma = 5.2 \), we'll need to use the standard normal distribution (also known as the \( Z \)-distribution). 

The standard normal variable \( Z \) is defined as:

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

This transformation converts \( X \) into a standard normal variable with mean 0 and standard deviation 1. We'll use the standard normal table (or a calculator) to find the probabilities corresponding to the calculated \( Z \)-scores.

### Part (a): Probability \( X < 55.2 \)

1. First, calculate the \( Z \)-score:
   \[
   Z = \frac{55.2 - 50}{5.2} = \frac{5.2}{5.2} = 1
   \]

2. Using the standard normal distribution table, the probability corresponding to \( Z = 1 \) is approximately 0.8413.

   **Thus, \( P(X < 55.2) \approx 0.8413 \).**

### Part (b): Probability \( X > 60.3 \)

1. Calculate the \( Z \)-score:
   \[
   Z = \frac{60.3 - 50}{5.2} = \frac{10.3}{5.2} \approx 1.98
   \]

2. The probability corresponding to \( Z = 1.98 \) is approximately 0.9761. Since we want \( P(X > 60.3) \), we take the complement:
   \[
   P(X > 60.3) = 1 - 0.9761 = 0.0239
   \]

   **Thus, \( P(X > 60.3) \approx 0.0239 \).**

### Part (c): Probability \( 52 < X < 57.2 \)

1. Calculate the \( Z \)-scores for \( X = 52 \) and \( X = 57.2 \):

   - For \( X = 52 \):
     \[
     Z = \frac{52 - 50}{5.2} = \frac{2}{5.2} \approx 0.385
     \]
     The corresponding probability is approximately 0.6517.

   - For \( X = 57.2 \):
     \[
     Z = \frac{57.2 - 50}{5.2} = \frac{7.2}{5.2} \approx 1.385
     \]
     The corresponding probability is approximately 0.9177.

2. The probability that \( X \) lies between 52 and 57.2 is the difference between these two probabilities:
   \[
   P(52 < X < 57.2) = 0.9177 - 0.6517 = 0.2660
   \]

   **Thus, \( P(52 < X < 57.2) \approx 0.2660 \).**

### Part (d): Probability \( 40 < X < 65 \)

1. Calculate the \( Z \)-scores for \( X = 40 \) and \( X = 65 \):

   - For \( X = 40 \):
     \[
     Z = \frac{40 - 50}{5.2} = \frac{-10}{5.2} \approx -1.923
     \]
     The corresponding probability is approximately 0.0274.

   - For \( X = 65 \):
     \[
     Z = \frac{65 - 50}{5.2} = \frac{15}{5.2} \approx 2.885
     \]
     The corresponding probability is approximately 0.9981.

2. The probability that \( X \) lies between 40 and 65 is:
   \[
   P(40 < X < 65) = 0.9981 - 0.0274 = 0.9707
   \]

   **Thus, \( P(40 < X < 65) \approx 0.9707 \).**

---

Would you like more details or have any questions? Here are some related questions:

1. What is the probability that \( X \) takes on a value less than 45?
2. How do we interpret the standard normal distribution table?
3. What is the \( Z \)-score if \( X \) is exactly equal to the mean \( \mu = 50 \)?
4. What does it mean if a \( Z \)-score is negative?
5. How do we calculate probabilities for normal distributions without using a table?

**Tip:** The area under the standard normal curve between any two \( Z \)-scores gives the probability that the random variable lies between those two values.

A random variable has a normal distribution with a mean of µ = 50 and the standard deviation σ = 5.2. What are the probabilities that the random variable will take on a value a. less than 55.2 b. greater than 60.3 c. between 52 and 57.2 d. between 40 and 65?

Solve probability problems for a random variable with a normal distribution (mean µ = 50, standard deviation σ = 5.2). Learn how to calculate the probabilities for values less than 55.2, greater than 60.3, and within specific ranges, using Z-scores and the standard normal distribution.

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